# 5.6E: Exercises for Section 5.6

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For exercises 1 - 8, compute each indefinite integral.

1) $$\displaystyle ∫e^{2x}\,dx$$

2) $$\displaystyle ∫e^{−3x}\,dx$$

$$\displaystyle ∫e^{−3x}\,dx \quad = \quad \frac{−1}{3}e^{−3x}+C$$

3) $$\displaystyle ∫2^x\,dx$$

4) $$\displaystyle ∫3^{−x}\,dx$$

$$\displaystyle ∫3^{−x}\,dx \quad = \quad −\frac{3^{−x}}{\ln 3}+C$$

5) $$\displaystyle ∫\frac{1}{2x}\,dx$$

6) $$\displaystyle ∫\frac{2}{x}\,dx$$

$$\displaystyle ∫\frac{2}{x}\,dx \quad = \quad 2\ln x+C \quad = \quad \ln(x^2)+C$$

7) $$\displaystyle ∫\frac{1}{x^2}\,dx$$

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

$$\displaystyle ∫\frac{1}{\sqrt{x}}\,dx \quad = \quad 2\sqrt{x}+C$$

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

9) $$\displaystyle ∫\frac{\ln x}{x}\,dx$$

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

$$\displaystyle ∫\frac{dx}{x(\ln x)^2} \quad = \quad −\frac{1}{\ln x}+C$$

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

12) $$\displaystyle ∫\frac{dx}{x\ln x\ln(\ln x)}$$

$$\displaystyle ∫\frac{dx}{x\ln x\ln(\ln x)} \quad = \quad \ln(\ln(\ln x))+C$$

13) $$\displaystyle ∫\tan θ\,dθ$$

14) $$\displaystyle ∫\frac{\cos x−x\sin x}{x\cos x}\,dx$$

$$\displaystyle ∫\frac{\cos x−x\sin x}{x\cos x}\,dx \quad = \quad \ln(x\cos x)+C$$

15) $$\displaystyle ∫\frac{\ln(\sin x)}{\tan x}\,dx$$

16) $$\displaystyle ∫\ln(\cos x)\tan x\,dx$$

$$\displaystyle ∫\ln(\cos x)\tan x\,dx \quad = \quad −\dfrac{1}{2}(\ln(\cos(x)))^2+C$$

17) $$\displaystyle ∫xe^{−x^2}\,dx$$

18) $$\displaystyle ∫x^2e^{−x^3}\,dx$$

$$\displaystyle ∫x^2e^{−x^3}\,dx \quad = \quad \dfrac{−e^{−x^3}}{3}+C$$

19) $$\displaystyle ∫e^{\sin x}\cos x\,dx$$

20) $$\displaystyle ∫e^{\tan x}\sec^2 x\,dx$$

$$\displaystyle ∫e^{\tan x}\sec^2 x\,dx\quad = \quad e^{\tan x}+C$$

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

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

$$\displaystyle ∫\frac{e^{\ln(1−t)}}{1−t}\,dt = \int \frac{1-t}{1-t}\,dt = \int 1\, dt \quad = \quad t+C$$

In exercises 23 - 28, verify by differentiation that $$\displaystyle ∫\ln x\,dx=x(\ln x−1)+C$$, then use appropriate changes of variables to compute the integral.

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

24) $$\displaystyle ∫x^2\ln^2 x\,dx$$

$$\displaystyle ∫x^2\ln^2 x\,dx \quad = \quad \dfrac{1}{9}x^3(\ln(x^3)−1)+C$$

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

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

$$\displaystyle ∫\frac{\ln x}{\sqrt{x}}\,dx \quad = \quad 2\sqrt{x}(\ln x−2)+C$$

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

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

$$\displaystyle ∫^{\ln x}_0e^t\,dt=e^t\bigg|^{\ln x}_0=e^{\ln x}−e^0=x−1$$

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

29) $$\displaystyle ∫\tan(2x)\,dx$$

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

$$\displaystyle ∫\frac{\sin(3x)−\cos(3x)}{\sin(3x)+\cos(3x)}\,dx \quad = \quad −\frac{1}{3}\ln|\sin(3x)+\cos(3x)| + C$$

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

32) $$\displaystyle ∫x\csc(x^2)\,dx$$

$$\displaystyle ∫x\csc(x^2)\,dx \quad = \quad −\frac{1}{2}\ln∣\csc(x^2)+\cot(x^2)∣+C$$

33) $$\displaystyle ∫\ln(\cos x)\tan x\,dx$$

34) $$\displaystyle ∫\ln(\csc x)\cot x\,dx$$

$$\displaystyle ∫\ln(\csc x)\cot x\,dx \quad = \quad −\frac{1}{2}(\ln(\csc x))^2+C$$

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

In exercises 36 - 40, evaluate the definite integral.

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

$$\displaystyle ∫^2_1\frac{1+2x+x^2}{3x+3x^2+x^3}\,dx \quad = \quad \frac{1}{3}\ln\left(\tfrac{26}{7}\right)$$

37) $$\displaystyle ∫^{π/4}_0\tan x\,dx$$

38) $$\displaystyle ∫^{π/3}_0\frac{\sin x−\cos x}{\sin x+\cos x}\,dx$$

$$\displaystyle ∫^{π/3}_0\frac{\sin x−\cos x}{\sin x+\cos x}\,dx \quad = \quad \ln(\sqrt{3}−1)$$

39) $$\displaystyle ∫^{π/2}_{π/6}\csc x\,dx$$

40) $$\displaystyle ∫^{π/3}_{π/4}\cot x\,dx$$

$$\displaystyle ∫^{π/3}_{π/4}\cot x\,dx \quad = \quad \frac{1}{2}\ln\frac{3}{2}$$

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

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

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

$$\displaystyle ∫\frac{y−1}{y+1}\,dy \quad = \quad y−2\ln|y+1|+C$$

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

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

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

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

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

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

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

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

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

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

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 $$[0,1]$$

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 $$[0,1]$$

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 $$[0,1]$$

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

$$\dfrac{11}{2}\ln 2$$

57) $$f(x)=2^{−x};\quad a=1,b=2$$

58) $$f(x)=2^{−x};\quad a=3,b=4$$

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

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 $$x=N+10$$ is, at most, $$0.01$$.

$$\displaystyle ∫^{N+1}_Nxe^{−x^2}\,dx=\frac{1}{2}(e^{−N^2}−e^{−(N+1)^2}).$$ 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 $$0<a≤b$$.

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

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 $$\displaystyle ∫^3_2x^x(1+\ln x)\,dx$$.

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 $$\ln\left(\dfrac{1}{x}\right)=−\ln x$$.

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 $$\displaystyle \ln x=∫^x_1\frac{dt}{x}$$ 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 $$E'(t)=E(t).$$

$$x=E(\ln(x)).$$ 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 $$\displaystyle ∫^1_{−1}\frac{1}{\sqrt{2π}}e^{−x^{2/2}}\,dx.$$

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