# 5.E: Integration (Exercises)

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These are homework exercises to accompany OpenStax's "Calculus" Textmap.

5.1: Approximating Areas

1.) State whether the given sums are equal or unequal.

1. $$\displaystyle \sum_{i=1}^{10}i$$ and $$\displaystyle \sum_{k=1}^{10}k$$
2. $$\displaystyle \sum_{i=1}^{10}i$$ and $$\displaystyle \sum_{i=6}^{15}(i−5)$$
3. $$\displaystyle \sum_{i=1}^{10}i(i−1)$$ and $$\displaystyle \sum_{j=0}^9(j+1)j$$
4. $$\displaystyle \sum_{i=1}^{10}i(i−1)$$ and $$\displaystyle \sum_{k=1}^{10}(k^2−k)$$

Solution: a. They are equal; both represent the sum of the first 10 whole numbers. b. They are equal; both represent the sum of the first 10 whole numbers. c. They are equal by substituting $$\displaystyle j=i−1.$$ d. They are equal; the first sum factors the terms of the second.

In the following exercises, use the rules for sums of powers of integers to compute the sums.

2) $$\displaystyle \sum_{i=5}^{10}i$$

3) $$\displaystyle \sum_{i=5}^{10}i^2$$

Solution: $$\displaystyle 385−30=355$$

Suppose that $$\displaystyle \sum_{i=1}^{100}a_i=15$$ and $$\displaystyle \sum_{i=1}^{100}b_i=−12.$$ In the following exercises, compute the sums.

4) $$\displaystyle \sum_{i=1}^{100}(a_i+b_i)$$

5) $$\displaystyle \sum_{i=1}^{100}(a_i−b_i)$$

Solution: $$\displaystyle 15−(−12)=27$$

6) $$\displaystyle \sum_{i=1}^{100}(3a_i−4b_i)$$

7) $$\displaystyle \sum_{i=1}^{100}(5a_i+4b_i)$$

Solution: $$\displaystyle 5(15)+4(−12)=27$$

In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.

8) $$\displaystyle \sum_{k=1}^{20}100(k^2−5k+1)$$

9) $$\displaystyle \sum_{j=1}^{50}(j^2−2j)$$

Solution: $$\displaystyle \sum_{j=1}^{50}j%2−2\sum_{j=1}^{50}j=\frac{(50)(51)(101)}{6}−\frac{2(50)(51)}{2}=40, 375$$

10) $$\displaystyle \sum_{j=11}^{20}(j^2−10j)$$

11) $$\displaystyle \sum_{k=1}^{25}[(2k)^2−100k]$$

Solution: $$\displaystyle 4\sum_{k=1}^{25}k^2−100\sum_{k=1}^{25}k=\frac{4(25)(26)(51)}{9}−50(25)(26)=−10, 400$$

Let $$\displaystyle L_n$$ denote the left-endpoint sum using n subintervals and let $$\displaystyle R_n$$ denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval.

12) $$\displaystyle L_4$$ for $$\displaystyle f(x)=\frac{1}{x−1}$$ on $$\displaystyle [2,3]$$

13) $$\displaystyle R_4$$ for $$\displaystyle g(x)=cos(πx)$$ on $$\displaystyle [0,1]$$

Solution: $$\displaystyle R_4=0.25$$

14) $$\displaystyle L_6$$ for $$\displaystyle f(x)=\frac{1}{x(x−1)}$$ on $$\displaystyle [2,5]$$

15) $$\displaystyle R_6$$ for $$\displaystyle f(x)=\frac{1}{x(x−1)}$$ on $$\displaystyle [2,5]$$

Solution: $$\displaystyle R_6=0.372$$

16) $$\displaystyle R_4$$ for $$\displaystyle \frac{1}{x^2+1}$$ on $$\displaystyle [−2,2]$$

17) $$\displaystyle L_4$$ for $$\displaystyle \frac{1}{x^2+1}$$ on $$\displaystyle [−2,2]$$

$$\displaystyle L_4=2.20$$

18) $$\displaystyle R_4$$ for $$\displaystyle x^2−2x+1$$ on $$\displaystyle [0,2]$$

19) $$\displaystyle L_8$$ for $$\displaystyle x^2−2x+1$$ on $$\displaystyle [0,2]$$

Solution: $$\displaystyle L_8=0.6875$$

20) Compute the left and right Riemann sums— $$\displaystyle L_4$$ and $$\displaystyle R_4$$, respectively—for $$\displaystyle f(x)=(2−|x|)$$ on $$\displaystyle [−2,2].$$ Compute their average value and compare it with the area under the graph of f.

21) Compute the left and right Riemann sums— $$\displaystyle L_6$$ and $$\displaystyle R_6$$, respectively—for $$\displaystyle f(x)=(3−|3−x|)$$ on $$\displaystyle [0,6].$$ Compute their average value and compare it with the area under the graph of f.

Solution: $$\displaystyle L_6=9.000=R_6$$. The graph of f is a triangle with area 9.

22) Compute the left and right Riemann sums— $$\displaystyle L_4$$ and $$\displaystyle R_4$$, respectively—for $$\displaystyle f(x)=\sqrt{4−x^2}$$ on $$\displaystyle [−2,2]$$ and compare their values.

23) Compute the left and right Riemann sums— $$\displaystyle L_6$$ and $$\displaystyle R_6$$, respectively—for $$\displaystyle f(x)=\sqrt{9−(x−3)^2}$$ on $$\displaystyle [0,6]$$ and compare their values.

Solution: $$\displaystyle L_6=13.12899=R_6$$. They are equal.

Express the following endpoint sums in sigma notation but do not evaluate them.

24) $$\displaystyle L_{30}$$ for $$\displaystyle f(x)=x^2$$ on $$\displaystyle [1,2]$$

25) $$\displaystyle L_{10}$$ for $$\displaystyle f(x)=\sqrt{4−x^2}$$ on $$\displaystyle [−2,2]$$

Solution: $$\displaystyle L_{10}=\frac{4}{10}\sum_{i=1}^{10}\sqrt{4−(−2+4\frac{(i−1)}{10})}$$

26) $$\displaystyle R_{20}$$ for $$\displaystyle f(x)=sinx$$ on $$\displaystyle [0,π]$$

27) $$\displaystyle R_{100}$$ for $$\displaystyle lnx$$ on $$\displaystyle [1,e]$$

Solution: $$\displaystyle R_{100}=\frac{e−1}{100}\sum_{i=1}^{100}ln(1+(e−1)\frac{i}{100})$$

In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and right endpoint sums?

28) [T] $$\displaystyle L_{100}$$ and $$\displaystyle R_{100}$$ for $$\displaystyle y=x^2−3x+1$$ on the interval $$\displaystyle [−1,1]$$

29) [T] $$\displaystyle L_{100}$$ and $$\displaystyle R_{100}$$ for $$\displaystyle y=x^2$$ on the interval $$\displaystyle [0,1]$$ $$\displaystyle R_{100}=0.33835,L_{100}=0.32835.$$ The plot shows that the left Riemann sum is an underestimate because the function is increasing. Similarly, the right Riemann sum is an overestimate. The area lies between the left and right Riemann sums. Ten rectangles are shown for visual clarity. This behavior persists for more rectangles.

30) [T] $$\displaystyle L_{50}$$ and $$\displaystyle R_{50}$$ for $$\displaystyle y=\frac{x+1}{x^2−1}$$ on the interval $$\displaystyle [2,4]$$

31) [T] $$\displaystyle L_{100}$$ and $$\displaystyle R_{100}$$ for $$\displaystyle y=x^3$$ on the interval $$\displaystyle [−1,1]$$ $$\displaystyle L_{100}=−0.02,R_{100}=0.02$$. The left endpoint sum is an underestimate because the function is increasing. Similarly, a right endpoint approximation is an overestimate. The area lies between the left and right endpoint estimates.

32) [T] $$\displaystyle L_{50}$$ and $$\displaystyle R_{50}$$ for $$\displaystyle y=tan(x)$$ on the interval $$\displaystyle [0,\frac{π}{4}]$$

33) [T] $$\displaystyle L_{100}$$ and $$\displaystyle R_{100}$$ for $$\displaystyle y=e^{2x}$$ on the interval $$\displaystyle [−1,1]$$ $$\displaystyle L_{100}=3.555,R_{100}=3.670$$. The plot shows that the left Riemann sum is an underestimate because the function is increasing. Ten rectangles are shown for visual clarity. This behavior persists for more rectangles.

34) Let $$\displaystyle t_j$$ denote the time that it took Tejay van Garteren to ride the jth stage of the Tour de France in 2014. If there were a total of 21 stages, interpret $$\displaystyle \sum_{j=1}^{21}t_j$$.

35) Let $$\displaystyle r_j$$ denote the total rainfall in Portland on the jth day of the year in 2009. Interpret $$\displaystyle \sum_{j=1}^{31}r_j$$.

Solution: The sum represents the cumulative rainfall in January 2009.

36) Let $$\displaystyle d_j$$ denote the hours of daylight and $$\displaystyle δ_j$$ denote the increase in the hours of daylight from day $$\displaystyle j−1$$ to day j in Fargo, North Dakota, on thejth day of the year. Interpret $$\displaystyle d1+\sum_{j=2}^{365}δ_j$$.

37) To help get in shape, Joe gets a new pair of running shoes. If Joe runs 1 mi each day in week 1 and adds $$\displaystyle \frac{1}{10}$$ mi to his daily routine each week, what is the total mileage on Joe’s shoes after 25 weeks?

Solution: The total mileage is $$\displaystyle 7×\sum_{i=1}^{25}(1+\frac{(i−1)}{10})=7×25+\frac{7}{10}×12×25=385mi$$.

38) The following table gives approximate values of the average annual atmospheric rate of increase in carbon dioxide (CO2) each decade since 1960, in parts per million (ppm). Estimate the total increase in atmospheric CO2 between 1964 and 2013.

1964-1973 1.07
1976-1983 1.34
1984-1993 1.40
1994-2003 1.87
2004-2013 2.07

Average Annual Atmospheric CO2 Increase, 1964–2013 Source: http://www.esrl.noaa.gov/gmd/ccgg/trends/.

39) The following table gives the approximate increase in sea level in inches over 20 years starting in the given year. Estimate the net change in mean sea level from 1870 to 2010.

Starting Year 20- Year Change
1870 0.3
1890 1.5
1910 0.2
1930 2.8
1950 0.7
1970 1.1
1990 1.5

Approximate 20-Year Sea Level Increases, 1870–1990

Solution: Add the numbers to get 8.1-in. net increase.

b. What percentage of households had incomes exceeding $85,000? c. Plot the data and try to fit its shape to that of a graph of the form $$\displaystyle a(x+c)e^{−b(x+e)}$$ for suitable $$\displaystyle a,b,c.$$ Solution: $$\displaystyle a. 54.3%; b. 27.00%; c.$$The curve in the following plot is $$\displaystyle 2.35(t+3)e^{−0.15(t+3)}.$$ ## 32) Newton’s law of gravity states that the gravitational force exerted by an object of mass M and one of mass m with centers that are separated by a distance r is $$\displaystyle F=G\frac{mM}{r^2}$$, with G an empirical constant $$\displaystyle G=6.67x10^{−11}m^3/(kg⋅s^2)$$. The work done by a variable force over an interval $$\displaystyle [a,b]$$ is defined as $$\displaystyle W=∫^b_aF(x)dx$$. If Earth has mass $$\displaystyle 5.97219×10^{24}$$ and radius 6371 km, compute the amount of work to elevate a polar weather satellite of mass 1400 kg to its orbiting altitude of 850 km above Earth. 33) For a given motor vehicle, the maximum achievable deceleration from braking is approximately 7 m/sec2 on dry concrete. On wet asphalt, it is approximately 2.5 m/sec2. Given that 1 mph corresponds to 0.447 m/sec, find the total distance that a car travels in meters on dry concrete after the brakes are applied until it comes to a complete stop if the initial velocity is 67 mph (30 m/sec) or if the initial braking velocity is 56 mph (25 m/sec). Find the corresponding distances if the surface is slippery wet asphalt. Solution: In dry conditions, with initial velocity $$\displaystyle v_0=30$$ m/s, $$\displaystyle D=64.3$$ and, if $$\displaystyle v_0=25,D=44.64$$. In wet conditions, if $$\displaystyle v_0=30$$, and $$\displaystyle D=180$$ and if $$\displaystyle v_0=25,D=125.$$ 34) John is a 25-year old man who weighs 160 lb. He burns $$\displaystyle 500−50t$$ calories/hr while riding his bike for t hours. If an oatmeal cookie has 55 cal and John eats 4t cookies during the tth hour, how many net calories has he lost after 3 hours riding his bike? 35) Sandra is a 25-year old woman who weighs 120 lb. She burns $$\displaystyle 300−50t$$ cal/hr while walking on her treadmill. Her caloric intake from drinking Gatorade is 100t calories during the tth hour. What is her net decrease in calories after walking for 3 hours? Solution: 225 cal 36) A motor vehicle has a maximum efficiency of 33 mpg at a cruising speed of 40 mph. The efficiency drops at a rate of 0.1 mpg/mph between 40 mph and 50 mph, and at a rate of 0.4 mpg/mph between 50 mph and 80 mph. What is the efficiency in miles per gallon if the car is cruising at 50 mph? What is the efficiency in miles per gallon if the car is cruising at 80 mph? If gasoline costs$3.50/gal, what is the cost of fuel to drive 50 mi at 40 mph, at 50 mph, and at 80 mph?

37) Although some engines are more efficient at given a horsepower than others, on average, fuel efficiency decreases with horsepower at a rate of $$\displaystyle 1/25$$ mpg/horsepower. If a typical 50-horsepower engine has an average fuel efficiency of 32 mpg, what is the average fuel efficiency of an engine with the following horsepower: 150, 300, 450?

Solution: $$\displaystyle E(150)=28,E(300)=22,E(450)=16$$

38) [T] The following table lists the 2013 schedule of federal income tax versus taxable income.

 Taxable Income Range The Tax Is ... ... Of the Amount Over $0–$8925 10% $0$8925–$36,250$892.50 + 15% $8925$36,250–$87,850$4,991.25 + 25% $36,250$87,850–$183,250$17,891.25 + 28% $87,850$183,250–$398,350$44,603.25 + 33% $183,250$398,350–$400,000$115,586.25 + 35% $398,350 >$400,000 $116,163.75 + 39.6%$400,000

Federal Income Tax Versus Taxable IncomeSource: http://www.irs.gov/pub/irs-prior/i1040tt--2013.pdf.

Suppose that Steve just received a $10,000 raise. How much of this raise is left after federal taxes if Steve’s salary before receiving the raise was$40,000? If it was $90,000? If it was$385,000?

39) [T] The following table provides hypothetical data regarding the level of service for a certain highway.

 Highway Speed Range (mph) Vehicles per Hour per Lane Density Range (vehicles/mi) >60 <600 <10 60-57 300-1000 10-20 57-54 1000-1500 20-30 57-54 1500-1900 30-45 46-30 1900-2100 48-70 <30 Unstable 70-200

a. Plot vehicles per hour per lane on the x-axis and highway speed on the y-axis.

b. Compute the average decrease in speed (in miles per hour) per unit increase in congestion (vehicles per hour per lane) as the latter increases from 600 to 1000, from 1000 to 1500, and from 1500 to 2100. Does the decrease in miles per hour depend linearly on the increase in vehicles per hour per lane?

c. Plot minutes per mile (60 times the reciprocal of miles per hour) as a function of vehicles per hour per lane. Is this function linear?

Solution:

a. b. Between 600 and 1000 the average decrease in vehicles per hour per lane is −0.0075. Between 1000 and 1500 it is −0.006 per vehicles per hour per lane, and between 1500 and 2100 it is −0.04 vehicles per hour per lane. c. The graph is nonlinear, with minutes per mile increasing dramatically as vehicles per hour per lane reach 2000.

40) For the next two exercises use the data in the following table, which displays bald eagle populations from 1963 to 2000 in the continental United States.

 Year Population of Breeding Pairs of Bald Eagles 1963 487 1974 791 1981 1188 1986 1875 1992 3749 1996 5094 2000 6471

Population of Breeding Bald Eagle PairsSource: http://www.fws.gov/Midwest/eagle/pop.../chtofprs.html.

41) [T] The graph below plots the quadratic $$\displaystyle p(t)=6.48t^2−80.31t+585.69$$ against the data in preceding table, normalized so that $$\displaystyle t=0$$ corresponds to 1963. Estimate the average number of bald eagles per year present for the 37 years by computing the average value of p over $$\displaystyle [0,37].$$ 42) [T] The graph below plots the cubic $$\displaystyle p(t)=0.07t^3+2.42t^2−25.63t+521.23$$ against the data in the preceding table, normalized so that $$\displaystyle t=0$$ corresponds to 1963. Estimate the average number of bald eagles per year present for the 37 years by computing the average value of p over $$\displaystyle [0,37].$$ Solution: $$\displaystyle \frac{1}{37}∫^{37}_0p(t)dt=\frac{0.07(37)^3}{4}+\frac{2.42(37)^2}{3}−\frac{25.63(37)}{2}+521.23≈2037$$

43) [T] Suppose you go on a road trip and record your speed at every half hour, as compiled in the following table. The best quadratic fit to the data is $$\displaystyle q(t)=5x^2−11x+49$$, shown in the accompanying graph. Integrate q to estimate the total distance driven over the 3 hours.

 Time (hr) Speed (m[h) 0 (start) 50 1 40 2 50 3 60

## As a car accelerates, it does not accelerate at a constant rate; rather, the acceleration is variable. For the following exercises, use the following table, which contains the acceleration measured at every second as a driver merges onto a freeway.

 Time (sec) Acceleration (mph/sex) 1 11.2 2 10.6 3 8.1 4 5.4 5 0

45) [T] The accompanying graph plots the best quadratic fit, $$\displaystyle a(t)=−0.70t^2+1.44t+10.44$$, to the data from the preceding table. Compute the average value of a(t) to estimate the average acceleration between $$\displaystyle t=0$$ and $$\displaystyle t=5.$$ Solution: Average acceleration is $$\displaystyle A=\frac{1}{5}∫^5_0a(t)dt=−\frac{0.7(5^2)}{3}+\frac{1.44(5)}{2}+10.44≈8.2$$ mph/s

46) [T] Using your acceleration equation from the previous exercise, find the corresponding velocity equation. Assuming the final velocity is 0 mph, find the velocity at time $$\displaystyle t=0.$$

47) [T] Using your velocity equation from the previous exercise, find the corresponding distance equation, assuming your initial distance is 0 mi. How far did you travel while you accelerated your car? (Hint: You will need to convert time units.)

Solution: $$\displaystyle d(t)=∫^1_0|v(t)|dt=∫^t_0(\frac{7}{30}t^3−0.72t^2−10.44t+41.033)dt=\frac{7}{120}t^4−0.24t^3−5.22t^3+41.033t.$$ Then, $$\displaystyle d(5)≈81.12 mph ×sec≈119$$ feet.

48) [T] The number of hamburgers sold at a restaurant throughout the day is given in the following table, with the accompanying graph plotting the best cubic fit to the data, $$\displaystyle b(t)=0.12t^3−2.13t^3+12.13t+3.91,$$ with $$\displaystyle t=0$$ corresponding to 9 a.m. and $$\displaystyle t=12$$ corresponding to 9 p.m. Compute the average value of $$\displaystyle b(t)$$ to estimate the average number of hamburgers sold per hour.

 Hours Past Midnight No. of Burgers Sold 9 3 12 28 15 20 18 30 21 45 49) [T] An athlete runs by a motion detector, which records her speed, as displayed in the following table. The best linear fit to this data, $$\displaystyle ℓ(t)=−0.068t+5.14$$, is shown in the accompanying graph. Use the average value of $$\displaystyle ℓ(t)$$ between $$\displaystyle t=$$0 and $$\displaystyle t=40$$ to estimate the runner’s average speed.

 Minutes Speed (m/sec) 0 5 10 4.8 20 3.6 30 3.0 40 2.5 Solution: $$\displaystyle \frac{1}{40} ∫^{40}_0(−0.068t+5.14)dt=−\frac{0.068(40)}{2}+5.14=3.78$$

## 5.5: Substitution

1) Why is u-substitution referred to as change of variable?

2) 2. If $$\displaystylef=g∘h$$, when reversing the chain rule, $$\displaystyle\frac{d}{d}x(g∘h)(x)=g′(h(x))h′(x)$$, should you take $$\displaystyleu=g(x)$$ or u= $$\displaystyleh(x)?$$

Solution: $$\displaystyleu=h(x)$$

In the following exercises, verify each identity using differentiation. Then, using the indicated u-substitution, identify f such that the integral takes the form $$\displaystyle∫f(u)du.$$

3) $$\displaystyle∫x\sqrt{x+1}x=\frac{2}{15}(x+1)^{3/2}(3x−2)+C;u=x+1$$

4) $$\displaystyle∫\frac{x^2}{\sqrt{x−1}}dx(x>1)=\frac{2}{15}\sqrt{x−1}(3x^2+4x+8)+C;u=x−1$$

Solution: $$\displaystylef(u)=\frac{(u+1)^2}{\sqrt{u}}$$

5) $$\displaystyle∫x\sqrt{4x^2+9}dx=\frac{1}{12}(4x^2+9)^{3/2}+C;u=4x^2+9$$

6) $$\displaystyle∫\frac{x}{\sqrt{4x^2+9}}dx=\frac{1}{4}\sqrt{4x^2+9}+C;u=4x^2+9$$

Solution: $$\displaystyledu=8xdx;f(u)=\frac{1}{8\sqrt{u}}$$

7) $$\displaystyle∫\frac{x}{(4x^2+9)^2}dx=−\frac{1}{8(4x^2+9)};u=4x^2+9$$

In the following exercises, find the antiderivative using the indicated substitution.

8) $$\displaystyle∫(x+1)^4dx;u=x+1$$

Solutio: $$\displaystyle\frac{1}{5}(x+1)^5+C$$

9) $$\displaystyle∫(x−1)^5dx;u=x−1$$

10) $$\displaystyle∫(2x−3)^{−7}dx;u=2x−3$$

Solution: $$\displaystyle−\frac{1}{12(3−2x)^6}+C$$

11) $$\displaystyle∫(3x−2)^{−11}dx;u=3x−2$$

12) $$\displaystyle∫\frac{x}{\sqrt{x^2+1}}dx;u=x^2+1$$

Solution: $$\displaystyle\sqrt{x^2+1}+C$$

13) $$\displaystyle∫\frac{x}{\sqrt{1−x^2}}dx;u=1−x^2$$

14) $$\displaystyle∫(x−1)(x^2−2x)^3dx;u=x^2−2x$$

Solution: $$\displaystyle\frac{1}{8}(x^2−2x)^4+C$$

15) $$\displaystyle∫(x^2−2x)(x^3−3x^2)^2dx;u=x^3=3x^2$$

16) $$\displaystyle∫cos^3θdθ;u=sinθ (Hint:cos^2θ=1−sin^2θ)$$

Solution: $$\sin θ−\frac{sin^3θ}{3}+C$$

17) $$\displaystyle∫sin^3θdθ;u=cosθ (Hint:sin^2θ=1−cos^2θ)$$

In the following exercises, use a suitable change of variables to determine the indefinite integral.

18) $$\displaystyle∫x(1−x)^{99}dx$$

Solution: $$\displaystyle\frac{(1−x)^{101}}{101}−\frac{(1−x)^{100}}{100}+C$$

19) $$\displaystyle∫t(1−t^2)^{10}dt$$

20) $$\displaystyle∫(11x−7)^{−3}dx$$

Solution: $$\displaystyle−\frac{1}{22(7−11x^2)}+C$$

21) $$\displaystyle∫(7x−11)^4dx$$

22) $$\displaystyle∫cos^3θsinθdθ$$

Solution: $$\displaystyle−\frac{cos^4θ}{4}+C$$

23) $$\displaystyle∫sin^7θcosθdθ$$

24) $$\displaystyle∫cos^2(πt)sin(πt)dt$$

Solution: $$\displaystyle−\frac{cos^3(πt)}{3π}+C$$

25) $$\displaystyle∫sin^2xcos^3xdx (Hint:sin^2x+cos^2x=1)$$

26) $$\displaystyle∫tsin(t^2)cos(t^2)dt$$

Solution: $$\displaystyle−\frac{1}{4}cos^2(t^2)+C$$

27) $$\displaystyle∫t^2cos^2(t^3)sin(t^3)dt$$

28) $$\displaystyle∫\frac{x^2}{(x^3−3)^2}dx$$

Solution: $$\displaystyle−\frac{1}{3(x^3−3)}+C$$

29) $$\displaystyle∫\frac{x^3}{\sqrt{1−x^2}}dx$$

30) $$\displaystyle∫\frac{y^5}{(1−y^3)^{3/2}}dy$$

Solution: $$\displaystyle−\frac{2(y^3−2)}{3\sqrt{1−y^3}}$$

31) $$\displaystyle∫cosθ(1−cosθ)^{99}sinθdθ$$

32) $$\displaystyle∫(1−cos^3θ)^{10}cos^2θsinθdθ$$

Solution: $$\displaystyle\frac{1}{33}(1−cos^3θ)^{11}+C$$

33) $$\displaystyle∫(cosθ−1)(cos^2θ−2cosθ)^3sinθdθ$$

34) $$\displaystyle∫(sin^2θ−2sinθ)(sin^3θ−3sin^2θ^)3cosθdθ$$

Solution: $$\displaystyle\frac{1}{12}(sin^3θ−3sin^2θ)^4+C$$

In the following exercises, use a calculator to estimate the area under the curve using left Riemann sums with 50 terms, then use substitution to solve for the exact answer.

35) [T] $$\displaystyley=3(1−x)^2$$ over $$\displaystyle[0,2]$$

36) [T] $$\displaystyley=x(1−x^2)^3$$ over $$\displaystyle[−1,2]$$

Solution: $$\displaystyleL_{50}=−8.5779.$$ The exact area is $$\displaystyle\frac{−81}{8}$$

37) [T] $$\displaystyley=sinx(1−cosx)^2$$ over $$\displaystyle[0,π]$$

38) [T] $$\displaystyley=\frac{x}{(x2^+1)^2}$$ over $$\displaystyle[−1,1]$$

Solution: $$\displaystyleL_{50}=−0.006399$$ … The exact area is 0.

In the following exercises, use a change of variables to evaluate the definite integral.

39) $$\displaystyle∫^1_0x\sqrt{1−x^2}dx$$

40) $$\displaystyle∫^1_0\frac{x}{\sqrt{1+x^2}}dx$$

Solution: $$\displaystyleu=1+x^2,du=2xdx,\frac{1}{2}∫^2_1u^{−1/2}du=\sqrt{2}−1$$

41) $$\displaystyle∫^2_0\frac{t}{\sqrt{5+t^2}}dt$$

42) $$\displaystyle∫^1_0\frac{t}{\sqrt{1+t^3}}dt$$

Solution: $$\displaystyleu=1+t^3,du=3t^2,\frac{1}{3}∫^2_1u^{−1/2}du=\frac{2}{3}(\sqrt{2}−1)$$

43) $$\displaystyle∫^{π/4}_0sec^2θtanθdθ$$

44) $$\displaystyle∫^{π/4}_0\frac{sinθ}{cos^4θ}dθ$$

Solution: $$\displaystyleu=cosθ,du=−sinθdθ,∫^1_{1/\sqrt{2}}u^{−4}du=\frac{1}{3}(2\sqrt{2}−1)$$

In the following exercises, evaluate the indefinite integral $$\displaystyle∫f(x)dx$$ with constant $$\displaystyleC=0$$ using u-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of C that would need to be added to the antiderivative to make it equal to the definite integral $$\displaystyleF(x)=∫^x_af(t)dt$$, with a the left endpoint of the given interval.

45) [T] $$\displaystyle∫(2x+1)e^{x^2+x−6}dx$$ over $$\displaystyle[−3,2]$$

46) [T] $$\displaystyle∫\frac{cos(ln(2x))}{x}dx$$ on $$\displaystyle[0,2]$$

Solution: The antiderivative is $$\displaystyley=sin(ln(2x))$$. Since the antiderivative is not continuous at $$\displaystylex=0$$, one cannot find a value of C that would make $$\displaystyley=sin(ln(2x))−C$$ work as a definite integral.

47) [T] $$\displaystyle∫\frac{3x^2+2x+1}{\sqrt{x^3+x^2+x+4}}dx$$ over $$\displaystyle[−1,2]$$

48) [T] $$\displaystyle∫\frac{sinx}{cos^3x}dx$$ over $$\displaystyle[−\frac{π}{3},\frac{π}{3}]$$ The antiderivative is $$\displaystyley=\frac{1}{2}sec^2x$$. You should take $$\displaystyleC=−2$$ so that $$\displaystyleF(−\frac{π}{3})=0.$$

49) [T] $$\displaystyle∫(x+2)e^{−x^2−4x+3}dx$$ over $$\displaystyle[−5,1]$$

50) [T] $$\displaystyle∫3x^2\sqrt{2x^3+1}dx$$ over $$\displaystyle[0,1]$$ The antiderivative is $$\displaystyley=\frac{1}{3}(2x^3+1)^{3/2}$$. One should take $$\displaystyleC=−\frac{1}{3}$$.

51) If $$\displaystyleh(a)=h(b)$$ in $$\displaystyle∫^b_ag'(h(x))h(x)dx,$$ what can you say about the value of the integral?

52) Is the substitution $$\displaystyleu=1−x^2$$ in the definite integral $$\displaystyle∫^2_0\frac{x}{1−x^2}dx$$ okay? If not, why not?

Solution: No, because the integrand is discontinuous at $$\displaystylex=1$$.

In the following exercises, use a change of variables to show that each definite integral is equal to zero.

53) $$\displaystyle∫^π_0cos^2(2θ)sin(2θ)dθ$$

54) $$\displaystyle∫^\sqrt{π}_0tcos(t^2)sin(t^2)dt$$

Solution: $$\displaystyleu=sin(t^2);$$ the integral becomes $$\displaystyle\frac{1}{2}∫^0_0udu.$$

55) $$\displaystyle∫^1_0(1−2t)dt$$

56) $$\displaystyle∫^1_0\frac{1−2t}{(1+(t−\frac{1}{2})^2)}dt$$

Solution: $$\displaystyleu=(1+(t−\frac{1}{2})^2);$$ the integral becomes $$\displaystyle−∫^{5/4}_{5/4}\frac{1}{u}du$$.

57) $$\displaystyle∫^π_0sin((t−\frac{π}{2})^3)cos(t−\frac{π}{2})dt$$

58) $$\displaystyle∫^2_0(1−t)cos(πt)dt$$

Solution: $$\displaystyleu=1−t;$$ the integral becomes

$$\displaystyle∫^{−1}_1ucos(π(1−u))du=∫^{−1}_1u[cosπcosu−sinπsinu]du=−∫^{−1}_1ucosudu=∫^{1−}_1ucosudu=0$$

since the integrand is odd.

59) $$\displaystyle∫^{3π/4}_{π/4}sin^2tcostdt$$

60) Show that the average value of $$\displaystylef(x)$$ over an interval $$\displaystyle[a,b]$$ is the same as the average value of $$\displaystylef(cx)$$ over the interval $$\displaystyle[\frac{a}{c},\frac{b}{c}]$$ for $$\displaystylec>0.$$

Solution: Setting $$\displaystyleu=cx$$ and $$\displaystyledu=cdx$$ gets you $$\displaystyle\frac{1}{\frac{b}{c}−\frac{a}{c}}∫^{b/c}_{a/c}f(cx)dx=\frac{c}{b−a}∫^{u=b}_{u=a}f(u)\frac{du}{c}=\frac{1}{b−a}∫^b_af(u)du.$$

61) Find the area under the graph of $$\displaystylef(t)=\frac{t}{(1+t^2)^a}$$ between $$\displaystylet=0$$ and $$\displaystylet=x$$ where $$\displaystylea>0$$ and $$\displaystylea≠1$$ is fixed, and evaluate the limit as $$\displaystylex→∞$$.

62) Find the area under the graph of $$\displaystyleg(t)=\frac{t}{(1−t^2)^a}$$ between $$\displaystylet=0$$ and $$\displaystylet=x$$, where $$\displaystyle0<x<1$$ and $$\displaystylea>0$$ is fixed. Evaluate the limit as $$\displaystylex→1$$.

Solution: $$\displaystyle∫^x_0g(t)dt=\frac{1}{2}∫^1_{u=1−x^2} \frac{du}{u^a}=\frac{1}{2(1−a)}u^{1−a}∣1u=\frac{1}{2(1−a)}(1−(1−x^2)^{1−a})$$ As $$\displaystylex→1$$ the limit is $$\displaystyle\frac{1}{2(1−a)}$$ if $$\displaystylea<1$$, and the limit diverges to +∞ if $$\displaystylea>1$$.

63) The area of a semicircle of radius 1 can be expressed as $$\displaystyle∫^1_{−1}\sqrt{1−x^2}dx$$. Use the substitution $$\displaystylex=cost$$ to express the area of a semicircle as the integral of a trigonometric function. You do not need to compute the integral.

64) The area of the top half of an ellipse with a major axis that is the x-axis from $$\displaystylex=−1$$ to a and with a minor axis that is the y-axis from $$\displaystyley=−b$$ to b can be written as $$\displaystyle∫^a_{−a}b\sqrt{1−\frac{x^2}{a^2}}dx$$. Use the substitution $$\displaystylex=acost$$ to express this area in terms of an integral of a trigonometric function. You do not need to compute the integral.

Solution: $$\displaystyle∫^0_{t=π}b\sqrt{1−cos^2t}×(−asint)dt=∫^π_t=0absin^2tdt$$

65) [T] The following graph is of a function of the form $$\displaystylef(t)=asin(nt)+bsin(mt)$$. Estimate the coefficients a and b, and the frequency parameters n and m. Use these estimates to approximate $$\displaystyle∫^π_0f(t)dt$$. 66) [T] The following graph is of a function of the form $$\displaystylef(x)=acos(nt)+bcos(mt)$$. Estimate the coefficients a and b and the frequency parameters n and m. Use these estimates to approximate $$\displaystyle∫^π_0f(t)dt.$$ Solution: $$\displaystylef(t)=2cos(3t)−cos(2t);∫^{π/2}_0(2cos(3t)−cos(2t))=−\frac{2}{3}$$

## 5.6: Integrals Involving Exponential and Logarithmic Functions

In the following exercises, compute each indefinite integral.

1) $$\displaystyle ∫e^{2x}dx$$

2) $$\displaystyle ∫e^{−3x}dx$$

Solution: $$\displaystyle \frac{−1}{3}e^{−3x}+C$$

3) $$\displaystyle ∫2^xdx$$

4) $$\displaystyle ∫3^{−x}dx$$

Solution: $$\displaystyle −\frac{3^{−x}}{ln3}+C$$

5) $$\displaystyle ∫\frac{1}{2x}dx$$

6) $$\displaystyle ∫\frac{2}{x}dx$$

Solution: $$\displaystyle ln(x^2)+C$$

7) $$\displaystyle ∫\frac{1}{x^2}dx$$

8) $$\displaystyle ∫\frac{1}{\sqrt{x}}dx$$

Soltuion: $$\displaystyle 2\sqrt{x}+C$$

In the following exercises, find each indefinite integral by using appropriate substitutions.

9) $$\displaystyle ∫\frac{lnx}{x}dx$$

10) $$\displaystyle ∫\frac{dx}{x(lnx)^2}$$

Solution: $$\displaystyle −\frac{1}{lnx}+C$$

11) $$\displaystyle ∫\frac{dx}{xlnx}(x>1)$$

12) $$\displaystyle ∫\frac{dx}{xlnxln(lnx)}$$

Solution: $$\displaystyle ln(ln(lnx))+C$$

13) $$\displaystyle ∫tanθdθ$$

14) $$\displaystyle ∫\frac{cosx−xsinx}{xcosx}dx$$

Solution: $$\displaystyle ln(xcosx)+C$$

15) $$\displaystyle ∫\frac{ln(sinx)}{tanx}dx$$

16) $$\displaystyle ∫ln(cosx)tanxdx$$

Solution: $$\displaystyle −\frac{1}{2}(ln(cos(x)))^2+C$$

17) $$\displaystyle ∫xe^{−x^2}dx$$

18) $$\displaystyle ∫x^2e^{−x^3}dx$$

Solution: $$\displaystyle \frac{−e^{−x^3}}{3}+C$$

19) $$\displaystyle ∫e^{sinx}cosxdx$$

20) $$\displaystyle ∫e^{tanx}sec^2xdx$$

Solution: $$\displaystyle e^{tanx}+C$$

21) $$\displaystyle ∫e^{lnx}\frac{dx}{x}$$

22) $$\displaystyle ∫\frac{e^{ln(1−t)}}{1−t}dt$$

Solution: $$\displaystyle t+C$$

In the following exercises, verify by differentiation that $$\displaystyle ∫lnxdx=x(lnx−1)+C$$, then use appropriate changes of variables to compute the integral.

23) $$\displaystyle ∫lnxdx (Hint:∫lnxdx=\frac{1}{2}∫xln(x^2)dx)$$

24) $$\displaystyle ∫x^2ln^2xdx$$

Solution: $$\displaystyle \frac{1}{9}x^3(ln(x^3)−1)+C$$

25) $$\displaystyle ∫\frac{lnx}{x^2}dx$$ (Hint:Set $$\displaystyle u=\frac{1}{x}.)$$

26) $$\displaystyle ∫\frac{lnx}{\sqrt{x}}dx$$ (Hint:Set $$\displaystyle u=\sqrt{x}.)$$

Solution: $$\displaystyle 2\sqrt{x}(lnx−2)+C$$

27) Write an integral to express the area under the graph of $$\displaystyle y=\frac{1}{t}$$ from $$\displaystyle t=1$$ to $$\displaystyle e^x$$ and evaluate the integral.

28) Write an integral to express the area under the graph of $$\displaystyle y=e^t$$ between $$\displaystyle t=0$$ and $$\displaystyle t=lnx$$, and evaluate the integral.

Solution: $$\displaystyle ∫^{lnx}_0e^tdt=e^t∣^{lnx}_0=e^{lnx}−e^0=x−1$$

In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms.

29) $$\displaystyle ∫tan(2x)dx$$

30) $$\displaystyle ∫\frac{sin(3x)−cos(3x)}{sin(3x)+cos(3x)}dx$$

Solution: $$\displaystyle −\frac{1}{3}ln(sin(3x)+cos(3x))$$

31) $$\displaystyle ∫\frac{xsin(x^2)}{cos(x^2)}dx$$

32) $$\displaystyle ∫xcsc(x^2)dx$$

Solution: $$\displaystyle −\frac{1}{2}ln∣csc(x^2)+cot(x^2)∣+C$$

33) $$\displaystyle ∫ln(cosx)tanxdx$$

34) $$\displaystyle ∫ln(cscx)cotxdx$$

Solution: $$\displaystyle −\frac{1}{2}(ln(cscx))^2+C$$

35) $$\displaystyle ∫\frac{e^x−e^{−x}}{e^x+e^{−x}}dx$$

In the following exercises, evaluate the definite integral.

36) $$\displaystyle ∫^2_1\frac{1+2x+x^2}{3x+3x^2+x^3}dx$$

Solution: $$\displaystyle \frac{1}{3}ln(\frac{26}{7})$$

37) $$\displaystyle ∫^{π/4}_0tanxdx$$

38) $$\displaystyle ∫^{π/3}_0\frac{sinx−cosx}{sinx+cosx}dx$$

Solutnio: $$\displaystyle ln(\sqrt{3}−1)$$

39) $$\displaystyle ∫^{π/2}_{π/6}cscxdx$$

40) $$\displaystyle ∫^{π/3}_{π/4}cotxdx$$

Solution: $$\displaystyle \frac{1}{2}ln\frac{3}{2}$$

In the following exercises, integrate using the indicated substitution.

41) $$\displaystyle ∫\frac{x}{x−100}dx;u=x−100$$

42) $$\displaystyle ∫\frac{y−1}{y+1}dy;u=y+1$$

Solution: $$\displaystyle y−2ln|y+1|+C$$

43) $$\displaystyle ∫\frac{1−x^2}{3x−x^3}dx;u=3x−x^3$$

44) $$\displaystyle ∫\frac{sinx+cosx}{sinx−cosx}dx;u=sinx−cosx$$

Solution: $$\displaystyle ln|sinx−cosx|+C$$

45) $$\displaystyle ∫e^{2x}\sqrt{1−e^{2x}}dx;u=e^{2x}$$

46) $$\displaystyle ∫ln(x)\frac{\sqrt{1−(lnx)^2}}{x}dx;u=lnx$$

Solution: $$\displaystyle −\frac{1}{3}(1−(lnx^2))^{3/2}+C$$

In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R50 and solve for the exact area.

47) [T] $$\displaystyle y=e^x$$ over $$\displaystyle [0,1]$$

48) [T] $$\displaystyle y=e^{−x}$$ over $$\displaystyle [0,1]$$

Solution: Exact solution: $$\displaystyle \frac{e−1}{e},R_{50}=0.6258$$. Since f is decreasing, the right endpoint estimate underestimates the area.

49) [T] $$\displaystyle y=ln(x)$$ over $$\displaystyle [1,2]$$

50) [T] $$\displaystyle y=\frac{x+1}{x^2+2x+6}$$ over $$\displaystyle [0,1]$$

Solution: Exact solution: $$\displaystyle \frac{2ln(3)−ln(6)}{2},R_{50}=0.2033.$$ Since f is increasing, the right endpoint estimate overestimates the area.

51) [T] $$\displaystyle y=2^x$$ over $$\displaystyle [−1,0]$$

52) [T] $$\displaystyle y=−2^{−x}$$over $$\displaystyle [0,1]$$

Solution: Exact solution: $$\displaystyle −\frac{1}{ln(4)},R_{50}=−0.7164.$$ Since f is increasing, the right endpoint estimate overestimates the area (the actual area is a larger negative number).

In the following exercises, $$\displaystyle f(x)≥0$$ for $$\displaystyle a≤x≤b$$. Find the area under the graph of $$\displaystyle f(x)$$ between the given values a and b by integrating.

53) $$\displaystyle f(x)=\frac{log_{10}(x)}{x};a=10,b=100$$

54) $$\displaystyle f(x)=\frac{log_2(x)}{x};a=32,b=64$$

Solution: $$\displaystyle \frac{11}{2}ln2$$

55) $$\displaystyle f(x)=2^{−x};a=1,b=2$$

56) $$\displaystyle f(x)=2^{−x};a=3,b=4$$

Solution: $$\displaystyle \frac{1}{ln(65,536)}$$

57) Find the area under the graph of the function $$\displaystyle f(x)=xe^{−x^2}$$ between $$\displaystyle x=0$$ and $$\displaystyle x=5$$.

58) Compute the integral of $$\displaystyle f(x)=xe^{−x^2}$$ and find the smallest value of N such that the area under the graph $$\displaystyle f(x)=xe^{−x^2}$$ between $$\displaystyle x=N$$ and $$\displaystyle x=N+10$$ is, at most, 0.01.

Solution: $$\displaystyle ∫^{N+1}_Nxe^{−x^2}dx=\frac{1}{2}(e^{−N^2}−e^{−(N+1)^2}).$$ The quantity is less than 0.01 when $$\displaystyle N=2$$.

59) Find the limit, as N tends to infinity, of the area under the graph of $$\displaystyle f(x)=xe^{−x^2}$$ between $$\displaystyle x=0$$ and $$\displaystyle x=5$$.

60) Show that $$\displaystyle ∫^b_a\frac{dt}{t}=∫^{1/a}_{1/b}\frac{dt}{t}$$ when $$\displaystyle 0<a≤b$$.

Solution: $$\displaystyle ∫^b_a\frac{dx}{x}=ln(b)−ln(a)=ln(\frac{1}{a})−ln(\frac{1}{b})=∫^{1/a}_{1/b}\frac{dx}{x}$$

61) Suppose that $$\displaystyle f(x)>0$$ for all x and that f and g are differentiable. Use the identity $$\displaystyle f^g=e^{glnf}$$ and the chain rule to find the derivative of $$\displaystyle f^g$$.

62) Use the previous exercise to find the antiderivative of $$\displaystyle h(x)=x^x(1+lnx)$$ and evaluate $$\displaystyle ∫^3_2x^x(1+lnx)dx$$.

Solution: 23

63) Show that if $$\displaystyle c>0$$, then the integral of $$\displaystyle 1/x$$ from ac to bc $$\displaystyle (0<a<b)$$ is the same as the integral of $$\displaystyle 1/x$$ from a to b.

The following exercises are intended to derive the fundamental properties of the natural log starting from the definition $$\displaystyle ln(x)=∫^x_1\frac{dt}{t}$$, using properties of the definite integral and making no further assumptions.

64) Use the identity $$\displaystyle ln(x)=∫^x_1\frac{dt}{t}$$ to derive the identity $$\displaystyle ln(\frac{1}{x})=−lnx$$.

Solution: We may assume that $$\displaystyle x>1$$,so $$\displaystyle \frac{1}{x}<1.$$ Then, $$\displaystyle ∫^{1/x}_{1}\frac{dt}{t}$$. Now make the substitution $$\displaystyle u=\frac{1}{t}$$, so $$\displaystyle du=−\frac{dt}{t^2}$$ and $$\displaystyle \frac{du}{u}=−\frac{dt}{t}$$, and change endpoints: $$\displaystyle ∫^{1/x}_1\frac{dt}{t}=−∫^x_1\frac{du}{u}=−lnx.$$

65) Use a change of variable in the integral $$\displaystyle ∫^{xy}_1\frac{1}{t}dt$$ to show that $$\displaystyle lnxy=lnx+lny$$ for $$\displaystyle x,y>0$$.

66) Use the identity $$\displaystyle lnx=∫^x_1\frac{dt}{x}$$ to show that $$\displaystyle ln(x)$$ is an increasing function of x on $$\displaystyle [0,∞)$$, and use the previous exercises to show that the range of $$\displaystyle ln(x)$$ is $$\displaystyle (−∞,∞)$$. Without any further assumptions, conclude that $$\displaystyle ln(x)$$ has an inverse function defined on $$\displaystyle (−∞,∞).$$

67) Pretend, for the moment, that we do not know that $$\displaystyle e^x$$ is the inverse function of $$\displaystyle ln(x)$$, but keep in mind that $$\displaystyle ln(x)$$ has an inverse function defined on $$\displaystyle (−∞,∞)$$. Call it E. Use the identity $$\displaystyle lnxy=lnx+lny$$ to deduce that $$\displaystyle E(a+b)=E(a)E(b)$$ for any real numbers a, b.

68) Pretend, for the moment, that we do not know that $$\displaystyle e^x$$ is the inverse function of $$\displaystyle lnx$$, but keep in mind that $$\displaystyle lnx$$ has an inverse function defined on $$\displaystyle (−∞,∞)$$. Call it E. Show that $$\displaystyle E'(t)=E(t).$$

Solution: $$\displaystyle x=E(ln(x)).$$ Then, $$\displaystyle 1=\frac{E'(lnx)}{x}$$ or $$\displaystyle x=E'(lnx)$$. Since any number t can be written $$\displaystyle t=lnx$$ for some x, and for such t we have $$\displaystyle x=E(t)$$, it follows that for any $$\displaystyle t,E'(t)=E(t).$$

69) The sine integral, defined as $$\displaystyle S(x)=∫^x_0\frac{sint}{t}dt$$ is an important quantity in engineering. Although it does not have a simple closed formula, it is possible to estimate its behavior for large x. Show that for $$\displaystyle k≥1,|S(2πk)−S(2π(k+1))|≤\frac{1}{k(2k+1)π}.$$ (Hint: $$\displaystyle sin(t+π)=−sint$$)

70) [T] The normal distribution in probability is given by $$\displaystyle p(x)=\frac{1}{σ\sqrt{2π}}e^{−(x−μ)^2/2σ^2}$$, where σ is the standard deviation and μ is the average. The standard normal distribution in probability, $$\displaystyle p_s$$, corresponds to $$\displaystyle μ=0$$ and $$\displaystyle σ=1$$. Compute the left endpoint estimates $$\displaystyle R_{10}$$ and $$\displaystyle R_{100}$$ of $$\displaystyle ∫^1_{−1}\frac{1}{\sqrt{2π}}e^{−x^{2/2}}dx.$$

Solution: $$\displaystyle R_{10}=0.6811,R_{100}=0.6827$$ 71) [T] Compute the right endpoint estimates $$\displaystyle R_{50}$$ and $$\displaystyle R_{100}$$ of $$\displaystyle ∫^5_{−3}\frac{1}{2\sqrt{2π}}e^{−(x−1)^2/8}$$.

## 5.7: Integrals Resulting in Inverse Trigonometric Functions

In the following exercises, evaluate each integral in terms of an inverse trigonometric function.

1) $$\displaystyle ∫^{\sqrt{3}/2}_0\frac{dx}{\sqrt{1−x^2}}$$

Solution: $$\displaystyle sin^{−1}x∣^{\sqrt{3}/2}_0=\frac{π}{3}$$

2) $$\displaystyle ∫^{1/2}_{−1/2}\frac{dx}{\sqrt{1−x^2}}$$

3) $$\displaystyle ∫^1_{\sqrt{3}}\frac{dx}{\sqrt{1+x^2}}$$

Solution: $$\displaystyle tan^{−1}x∣^1_{\sqrt{3}}=−\frac{π}{12}$$

4) $$\displaystyle ∫^{\sqrt{3}}_{\frac{1}{\sqrt{3}}}\frac{dx}{1+x^2}$$

5) $$\displaystyle ∫^{\sqrt{2}}_1\frac{dx}{|x|\sqrt{x^2−1}}$$

Solution: $$\displaystyle sec^{−1}x∣^{\sqrt{2}}_1=\frac{π}{4}$$

6) $$\displaystyle ∫^{\frac{2}{\sqrt{3}}}_1\frac{dx}{|x|\sqrt{x^2−1}}$$

In the following exercises, find each indefinite integral, using appropriate substitutions.

7) $$\displaystyle ∫\frac{dx}{\sqrt{9−x^2}}$$

Solution: $$\displaystyle sin^{−1}(\frac{x}{3})+C$$

8) $$\displaystyle ∫\frac{dx}{\sqrt{1−16x^2}}$$

9) $$\displaystyle ∫\frac{dx}{9+x^2}$$

Solution: $$\displaystyle \frac{1}{3}tan^{−1}(\frac{x}{3})+C$$

10) $$\displaystyle ∫\frac{dx}{25+16x^2}$$

11) $$\displaystyle ∫\frac{dx}{|x|\sqrt{x^2−9}}$$

Solution: $$\displaystyle \frac{1}{3}sec^{−1}(\frac{x}{3})+C$$

12) $$\displaystyle ∫\frac{dx}{|x|\sqrt{4x^2−16}}$$

13) Explain the relationship $$\displaystyle −cos^{−}1t+C=∫\frac{dt}{\sqrt{1−t^2}}=sin^{−1}t+C.$$ Is it true, in general, that $$\displaystyle cos^{−1}t=−sin^{−1}t$$?

Solution: $$\displaystyle cos(\frac{π}{2}−θ)=sinθ.$$ So, $$\displaystyle sin^{−1}t=\frac{π}{2}−cos^{−1}t.$$ They differ by a constant.

14) Explain the relationship $$\displaystyle sec^{−1}t+C=∫\frac{dt}{|t|sqrt{t^2−1}}=−csc^{−1}t+C.$$ Is it true, in general, that $$\displaystyle sec^{−1}t=−csc^{−1}t$$?

15) Explain what is wrong with the following integral: $$\displaystyle ∫^2_1\frac{dt}{\sqrt{1−t^2}}$$.

Soution: $$\displaystyle \sqrt{1−t^2}$$ is not defined as a real number when $$\displaystyle t>1$$.

16) Explain what is wrong with the following integral: $$\displaystyle ∫^1_{−1}\frac{dt}{|t|\sqrt{t^2−1}}$$.

In the following exercises, solve for the antiderivative $$\displaystyle ∫f$$ of f with $$\displaystyle C=0$$, then use a calculator to graph f and the antiderivative over the given interval $$\displaystyle [a,b]$$. Identify a value of C such that adding C to the antiderivative recovers the definite integral $$\displaystyle F(x)=∫^x_af(t)dt$$.

17) [T] $$\displaystyle ∫\frac{1}{\sqrt{9−x^2}}dx$$ over $$\displaystyle [−3,3]$$

Solution: The antiderivative is $$\displaystyle sin^{−1}(\frac{x}{3})+C$$. Taking $$\displaystyle C=\frac{π}{2}$$ recovers the definite integral.

18) [T] $$\displaystyle ∫\frac{9}{9+x^2}dx$$ over $$\displaystyle [−6,6]$$

19) [T] $$\displaystyle ∫\frac{cosx}{4+sin^2x}dx$$ over $$\displaystyle [−6,6]$$

Solution: The antiderivative is $$\displaystyle \frac{1}{2}tan^{−1}(\frac{sinx}{2})+C$$. Taking $$\displaystyle C=\frac{1}{2}tan^{−1}(\frac{sin(6)}{2})$$ recovers the definite integral.

20) [T] $$\displaystyle ∫\frac{e^x}{1+e^{2x}}dx$$ over $$\displaystyle [−6,6]$$

In the following exercises, compute the antiderivative using appropriate substitutions.

21) $$\displaystyle ∫\frac{sin^{−1}tdt}{\sqrt{1−t^2}}$$

Solution: $$\displaystyle \frac{1}{2}(sin^{−1}t)^2+C$$

22) $$\displaystyle ∫\frac{dt}{sin^{−1}\sqrt{t^1−t^2}}$$

23) $$\displaystyle ∫\frac{tan^{−1}(2t)1}+4t^2dt$$

Soluion: \frac{1}{4}(tan^{−1}(2t))^2\)

24\) ∫\frac{ttan^{−1}{t^2}1+t4dt

25) $$\displaystyle ∫sec^{−1}\frac{(}{t}2)|t|\sqrt{t^2−4}dt$$

Solution:L $$\displaystyle \frac{1}{4}(sec^{−1}(\frac{t}{2})2)+$$

26) $$\displaystyle ∫\frac{tsec^{−1}(t^2)}{t^2\sqrt{t^4−1}}dt$$

In the following exercises, use a calculator to graph the antiderivative $$\displaystyle ∫f$$ with $$\displaystyle C=0$$ over the given interval $$\displaystyle [a,b].$$ Approximate a value of C, if possible, such that adding C to the antiderivative gives the same value as the definite integral $$\displaystyle F(x)=∫^x_af(t)dt.$$

27) [T] $$\displaystyle ∫\frac{1}{x\sqrt{x^2−4}}dx$$ over $$\displaystyle [2,6]$$

Solution: The antiderivative is $$\displaystyle \frac{1}{2}sec^{−1}(\frac{x}{2})+C$$. Taking $$\displaystyle C=0$$ recovers the definite integral over $$\displaystyle [2,6]$$.

28) [T] $$\displaystyle ∫\frac{1}{(2x+2)\sqrt{x}}dx$$ over $$\displaystyle [0,6]$$

29) [T] $$\displaystyle ∫\frac{(sinx+xcosx)}{1+x^2sin^2xdx}$$ over $$\displaystyle [−6,6]$$

Solution: The general antiderivative is $$\displaystyle tan^{−1}(xsinx)+C$$. Taking $$\displaystyle C=−tan^{−1}(6sin(6))$$ recovers the definite integral.

30) [T] $$\displaystyle ∫\frac{2e^{−2x}}{\sqrt{1−e^{−4x}}}dx$$ over $$\displaystyle [0,2]$$

31) [T] $$\displaystyle ∫\frac{1}{x+xln2x}$$ over $$\displaystyle [0,2]$$ The general antiderivative is $$\displaystyle tan^{−1}(lnx)+C$$. Taking $$\displaystyle C=\frac{π}{2}=tan^{−1}∞$$ recovers the definite integral.

32) [T] $$\displaystyle ∫\frac{sin^{−1}x}{\sqrt{1−x^2}}$$ over $$\displaystyle [−1,1]$$

In the following exercises, compute each integral using appropriate substitutions.

33) $$\displaystyle ∫\frac{e^x}{\sqrt{1−e^{2t}}}dt$$

Solutnio: $$\displaystyle sin^{−1}(e^t)+C$$

34) $$\displaystyle ∫\frac{e^t}{1+e^{2t}}dt$$

35) $$\displaystyle ∫\frac{dt}{t\sqrt{1−ln^2t}}$$

Solution: $$\displaystyle sin^{−1}(lnt)+C$$

36) $$\displaystyle ∫\frac{dt}{t(1+ln^2t)}$$

37) $$\displaystyle ∫\frac{cos^{−1}(2t)}{\sqrt{1−4t^2}}dt$$

Solution: $$\displaystyle −\frac{1}{2}(cos^{−1}(2t))^2+C$$

38) $$\displaystyle ∫\frac{e^tcos^{−1}(e^t)}{\sqrt{1−e^{2t}}}dt$$

In the following exercises, compute each definite integral.

39) $$\displaystyle ∫^{1/2}|_0\frac{tan(sin^{−1}t)}{\sqrt{1−t^2}}dt$$

Solution: $$\displaystyle \frac{1}{2}ln(\frac{4}{3})$$

40) $$\displaystyle ∫^{1/2}_{1/4}\frac{tan(cos^{−1}t)}{\sqrt{1−t^2}}dt$$

41) $$\displaystyle ∫^{1/2}_0\frac{sin(tan^{−1}t)}{1+t^2}dt$$

Solution: $$\displaystyle 1−\frac{2}{\sqrt{5}}$$

42) $$\displaystyle ∫^{1/2}_0\frac{cos(tan^{−1}t)}{1+t^2}dt\0 43) For \(\displaystyle A>0$$, compute $$\displaystyle I(A)=∫^{A}_{−A}\frac{dt}{1+t^2}$$ and evaluate $$\displaystyle \lim_{a→∞}I(A)$$, the area under the graph of $$\displaystyle \frac{1}{1+t^2}$$ on $$\displaystyle [−∞,∞]$$.

Solution: $$\displaystyle 2tan^{−1}(A)→π$$ as $$\displaystyle A→∞$$

44) For $$\displaystyle 1<B<∞$$, compute $$\displaystyle I(B)=∫^B_1\frac{dt}{t\sqrt{t^2−1}}$$ and evaluate $$\displaystyle \lim_{B→∞}I(B)$$, the area under the graph of $$\displaystyle \frac{1}{t\sqrt{t^2−1}}$$ over $$\displaystyle [1,∞)$$.

45) Use the substitution $$\displaystyle u=\sqrt{2}cotx$$ and the identity 1 $$\displaystyle +cot^2x=csc^2x$$ to evaluate $$\displaystyle ∫\frac{dx}{1+cos^2x}$$. (Hint: Multiply the top and bottom of the integrand by $$\displaystyle csc^2x$$.)

Solution: Using the hint, one has $$\displaystyle ∫\frac{csc^2x}{csc^2x+cot^2x}dx=∫\frac{csc^2x}{1+2cot^2x}dx.$$ Set $$\displaystyle u=\sqrt{2}cotx.$$ Then, $$\displaystyle du=−\sqrt{2}csc^2x$$ and the integral is $$\displaystyle −\frac{1}{\sqrt{2}}∫\frac{du}{1+u^2}=−\frac{1}{\sqrt{2}}tan^{−1}u+C=\frac{1}{\sqrt{2}}tan^{−1}(\sqrt{2}cotx)+C$$. If one uses the identity $$\displaystyle tan^{−1}s+tan^{−1}(\frac{1}{s})=\frac{π}{2}$$, then this can also be written $$\displaystyle \frac{1}{\sqrt{2}}tan^{−1}(\frac{tanx}{\sqrt{2}})+C.$$

46) [T] Approximate the points at which the graphs of $$\displaystyle f(x)=2x^2−1$$ and $$\displaystyle g(x)=(1+4x^2)^{−3/2}$$ intersect, and approximate the area between their graphs accurate to three decimal places.

47) [T] Approximate the points at which the graphs of $$\displaystyle f(x)=x^2−1$$ and $$\displaystyle f(x)=x^2−1$$ intersect, and approximate the area between their graphs accurate to three decimal places.

Solution: $$\displaystyle x≈±1.13525.$$ The left endpoint estimate with $$\displaystyle N=100$$ is 2.796 and these decimals persist for $$\displaystyle N=500$$.

48) Use the following graph to prove that $$\displaystyle ∫^x_0\sqrt{1−t^2}dt=\frac{1}{2}x\sqrt{1−x^2}+\frac{1}{2}sin^{−1}x.$$ ## Chapter Review Exercises:

True or False. Justify your answer with a proof or a counterexample. Assume all functions $$\displaystyle f$$ and $$\displaystyle g$$ are continuous over their domains.

1) If $$\displaystyle f(x)>0,f′(x)>0$$ for all $$\displaystyle x$$, then the right-hand rule underestimates the integral $$\displaystyle ∫^b_af(x).$$ Use a graph to justify your answer.

Solution: False

2) $$\displaystyle ∫^b_af(x)^2dx=∫^b_af(x)dx∫^b_af(x)dx$$

3) If $$\displaystyle f(x)≤g(x)$$ for all $$\displaystyle x∈[a,b]$$, then $$\displaystyle ∫^b_af(x)≤∫^b_ag(x).$$

Solution: True

4) All continuous functions have an antiderivative.

Evaluate the Riemann sums $$\displaystyle L_4$$ and $$\displaystyle R_4$$ for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer.

5) $$\displaystyle y=3x^2−2x+1)$$ over $$\displaystyle [−1,1]$$

Solution: $$\displaystyle L_4=5.25,R_4=3.25,$$ exact answer: 4

6) $$\displaystyle y=ln(x^2+1)$$ over $$\displaystyle [0,e]$$

7) $$\displaystyle y=x^2sinx$$ over $$\displaystyle [0,π]$$

Solution: $$\displaystyle L_4=5.364,R_4=5.364,$$ exact answer: $$\displaystyle 5.870$$

8) $$\displaystyle y=\sqrt{x}+\frac{1}{x}$$ over $$\displaystyle [1,4]$$

Evaluate the following integrals.

9) $$\displaystyle ∫^1_{−1}(x^3−2x^2+4x)dx$$

Solution: $$\displaystyle −\frac{4}{3}$$

10) $$\displaystyle ∫^4_0\frac{3t}{\sqrt{1+6t^2}}dt$$

11) $$\displaystyle ∫^{π/2}_{π/3}2sec(2θ)tan(2θ)dθ$$

Solution: 1

12) $$\displaystyle ∫^{π/4}_0e^{cos^2x}sinxcosdx$$

Find the antiderivative.

13) $$\displaystyle ∫\frac{dx}{(x+4)^3}$$

Solution: $$\displaystyle −\frac{1}{2(x+4)^2}+C$$

14) $$\displaystyle ∫xln(x^2)dx$$

15) $$\displaystyle ∫\frac{4x^2}{\sqrt{1−x^6}}dx$$

Solution: $$\displaystyle \frac{4}{3}sin^{−1}(x^3)+C$$

16) $$\displaystyle ∫\frac{e^{2x}}{1+e^{4x}}dx$$

Find the derivative.

17) $$\displaystyle \frac{d}{dt}∫^t_0\frac{sinx}{\sqrt{1+x^2}}dx$$

Solution: $$\displaystyle \frac{sint}{\sqrt{1+t^2}}$$

18) $$\displaystyle \frac{d}{dx}∫^{x^3}_1\sqrt{4−t^2}dt$$

19) $$\displaystyle \frac{d}{dx}∫^{ln(x)}_1(4t+e^t)dt$$

Solution: $$\displaystyle 4\frac{lnx}{x}+1$$

20) $$\displaystyle \frac{d}{dx}∫^{cosx}_0e^{t^2}dt$$

The following problems consider the historic average cost per gigabyte of RAM on a computer.

 Year 5-Year Change ($) 1980 0 1985 −5,468,750 1990 −755,495 1995 −73,005 2000 −29,768 2005 −918 2010 −177 21) If the average cost per gigabyte of RAM in 2010 is$12, find the average cost per gigabyte of RAM in 1980.

Solution: $6,328,113 22) The average cost per gigabyte of RAM can be approximated by the function $$\displaystyle C(t)=8,500,000(0.65)^t$$, where $$\displaystyle t$$ is measured in years since 1980, and $$\displaystyle C$$ is cost in US$. Find the average cost per gigabyte of RAM for 1980 to 2010.

23) Find the average cost of 1GB RAM for 2005 to 2010.

Solution: \$73.36

24) The velocity of a bullet from a rifle can be approximated by $$\displaystyle v(t)=6400t^2−6505t+2686,$$ where $$\displaystyle t$$ is seconds after the shot and v is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: $$\displaystyle 0≤t≤0.5.$$ What is the total distance the bullet travels in 0.5 sec?

25) What is the average velocity of the bullet for the first half-second?

Solution: $$\displaystyle \frac{19117}{12}ft/sec,or1593ft/sec$$

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