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Mathematics LibreTexts

5.6E: Exercises for Section 5.6

( \newcommand{\kernel}{\mathrm{null}\,}\)

For exercises 1 - 8, compute each indefinite integral.

1) e2xdx

2) e3xdx

Answer
e3xdx=13e3x+C

3) 2xdx

4) 3xdx

Answer
3xdx=3xln3+C

5) 12xdx

6) 2xdx

Answer
2xdx=2ln|x|+C=ln(x2)+C

7) 1x2dx

8) 1xdx

Answer
1xdx=2x+C

In exercises 9 - 16, find each indefinite integral by using appropriate substitutions.

9) lnxxdx

10) dxx(lnx)2

Answer
dxx(lnx)2=1lnx+C

11) dxxlnx(x>1)

12) dxxlnxln(lnx)

Answer
dxxlnxln(lnx)=ln(ln(lnx))+C

13) tanθdθ

14) cosxxsinxxcosxdx

Answer
cosxxsinxxcosxdx=ln(xcosx)+C

15) ln(sinx)tanxdx

16) ln(cosx)tanxdx

Answer
ln(cosx)tanxdx=12(ln(cos(x)))2+C

17) xex2dx

18) x2ex3dx

Answer
x2ex3dx=ex33+C

19) esinxcosxdx

20) etanxsec2xdx

Answer
etanxsec2xdx=etanx+C

21) elnxxdx

22) eln(1t)1tdt

Answer
eln(1t)1tdt=1t1tdt=1dt=t+C

In exercises 23 - 28, verify by differentiation that lnxdx=x(lnx1)+C, then use appropriate changes of variables to compute the integral.

23) lnxdx (Hint: lnxdx=12xln(x2)dx)

24) x2ln2xdx

Answer
x2ln2xdx=19x3(ln(x3)1)+C

25) lnxx2dx (Hint: Set u=1x.)

26) lnxxdx (Hint: Set u=x.)

Answer
lnxxdx=2x(lnx2)+C

27) Write an integral to express the area under the graph of y=1t from t=1 to ex and evaluate the integral.

28) Write an integral to express the area under the graph of y=et between t=0 and t=lnx, and evaluate the integral.

Answer
lnx0etdt=et|lnx0=elnxe0=x1

In exercises 29 - 35, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms.

29) tan(2x)dx

30) sin(3x)cos(3x)sin(3x)+cos(3x)dx

Answer
sin(3x)cos(3x)sin(3x)+cos(3x)dx=13ln|sin(3x)+cos(3x)|+C

31) xsin(x2)cos(x2)dx

32) xcsc(x2)dx

Answer
xcsc(x2)dx=12lncsc(x2)+cot(x2)+C

33) ln(cosx)tanxdx

34) ln(cscx)cotxdx

Answer
ln(cscx)cotxdx=12(ln(cscx))2+C

35) exexex+exdx

In exercises 36 - 40, evaluate the definite integral.

36) 211+2x+x23x+3x2+x3dx

Answer
211+2x+x23x+3x2+x3dx=13ln(267)

37) π/40tanxdx

38) π/30sinxcosxsinx+cosxdx

Answer
π/30sinxcosxsinx+cosxdx=ln(31)

39) π/2π/6cscxdx

40) π/3π/4cotxdx

Answer
π/3π/4cotxdx=12ln32

In exercises 41 - 46, integrate using the indicated substitution.

41) xx100dx;u=x100

42) y1y+1dy;u=y+1

Answer
y1y+1dy=y2ln|y+1|+C

43) 1x23xx3dx;u=3xx3

44) \displaystyle ∫\frac{\sin x+\cos x}{\sin x−\cos x}\,dx;\quad u=\sin x−\cos x

Answer
\displaystyle ∫\frac{\sin x+\cos x}{\sin x−\cos x}\,dx \quad=\quad \ln|\sin x−\cos x|+C

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

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

Answer
\displaystyle ∫\ln(x)\frac{\sqrt{1−(\ln x)^2}}{x}\,dx \quad = \quad −\frac{1}{3}(1−(\ln x)^2)^{3/2}+C

47) \displaystyle \int \frac{\sqrt{x}}{\sqrt{x} + 2}\,dx; \quad u = \sqrt{x} + 2

Answer
\displaystyle \int \frac{\sqrt{x}}{\sqrt{x} + 2}\,dx \quad = \quad \left( \sqrt{x} + 2 \right)^2 - 8\left( \sqrt{x} + 2 \right) + 8\ln\left( \sqrt{x} + 2 \right) + C

48) \displaystyle \int e^x\sec(e^x+1)\tan(e^x+1)\,dx; \quad u = e^{x} + 1

Answer
\displaystyle \int e^x\sec(e^x+1)\tan(e^x+1)\,dx \quad = \quad \sec(e^x+1) + C

In exercises 49 - 54, state whether the right-endpoint approximation overestimates or underestimates the exact area. Then calculate the right endpoint estimate R_{50} and solve for the exact area.

49) [T] y=e^x over [0,1]

50) [T] y=e^{−x} over [0,1]

Answer
Since f is decreasing, the right endpoint estimate underestimates the area.
Exact solution: \dfrac{e−1}{e},\quad R_{50}=0.6258.

51) [T] y=\ln(x) over [1,2]

52) [T] y=\dfrac{x+1}{x^2+2x+6} over [0,1]

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

53) [T] y=2^x over [−1,0]

54) [T] y=−2^{−x} over [0,1]

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

In exercises 55 - 58, f(x)≥0 for a≤x≤b. Find the area under the graph of f(x) between the given values a and b by integrating.

55) f(x)=\dfrac{\log_{10}(x)}{x};\quad a=10,b=100

56) f(x)=\dfrac{\log_2(x)}{x};\quad a=32,b=64

Answer
\dfrac{11}{2}\ln 2

57) f(x)=2^{−x};\quad a=1,b=2

58) f(x)=2^{−x};\quad a=3,b=4

Answer
\dfrac{1}{\ln(65,536)}

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

60) Compute the integral of f(x)=xe^{−x^2} and find the smallest value of N such that the area under the graph f(x)=xe^{−x^2} between x=N and x=N+1 is, at most, 0.01.

Answer
\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 N=2.

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

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

Answer
\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}

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

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

Answer
23

65) Show that if c>0, then the integral of \frac{1}{x} from ac to bc (\text{for}\,0<a<b) is the same as the integral of \frac{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.

66) Use the identity \displaystyle \ln(x)=∫^x_1\frac{dt}{t} to derive the identity \ln\left(\dfrac{1}{x}\right)=−\ln x.

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

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

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

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

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

Answer
x=E(\ln(x)). Then, 1=\dfrac{E'(\ln x)}{x} or x=E'(\ln x). Since any number t can be written t=\ln x for some x, and for such t we have x=E(t), it follows that for any t,\,E'(t)=E(t).

71) The sine integral, defined as \displaystyle S(x)=∫^x_0\frac{\sin t}{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 k≥1,\quad |S(2πk)−S(2π(k+1))|≤\dfrac{1}{k(2k+1)π}. (Hint: \sin(t+π)=−\sin t)

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

Answer
R_{10}=0.6811,\quad R_{100}=0.6827

A graph of the function f(x) = .5 * ( sqrt(2)*e^(-.5x^2)) / sqrt(pi). It is a downward opening curve that is symmetric across the y axis, crossing at about (0, .4). It approaches 0 as x goes to positive and negative infinity. Between 1 and -1, ten rectangles are drawn for a right endpoint estimate of the area under the curve.

73) [T] Compute the right endpoint estimates R_{50} and R_{100} of \displaystyle ∫^5_{−3}\frac{1}{2\sqrt{2π}}e^{−(x−1)^2/8}.

  • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

  • Paul Seeburger (Monroe Community College) added problems 47-48 to Section 5.6 exercises.

5.6E: Exercises for Section 5.6 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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