Learning Objectives
- Use substitution to evaluate definite integrals.
Substitution for Definite Integrals
Substitution can be used with definite integrals, too. However, using substitution to evaluate a definite integral requires a change to the limits of integration. If we change variables in the integrand, the limits of integration change as well.
Substitution with Definite Integrals
Let [latex]u=g(x)[/latex] and let [latex]{g}^{\text{′}}[/latex] be continuous over an interval [latex]\left[a,b\right],[/latex] and let [latex]f[/latex] be continuous over the range of [latex]u=g(x).[/latex] Then,
Although we will not formally prove this theorem, we justify it with some calculations here. From the substitution rule for indefinite integrals, if [latex]F(x)[/latex] is an antiderivative of [latex]f(x),[/latex] we have
Then
and we have the desired result.
Using Substitution to Evaluate a Definite Integral
Use substitution to evaluate [latex]{\int }_{0}^{1}{x}^{2}{(1+2{x}^{3})}^{5}dx.[/latex]
Use substitution to evaluate the definite integral [latex]{\int }_{-1}^{0}y{(2{y}^{2}-3)}^{5}dy.[/latex]
Using Substitution with an Exponential Function
Use substitution to evaluate [latex]{\int }_{0}^{1}x{e}^{4{x}^{2}+3}dx.[/latex]
Use substitution to evaluate [latex]{\int }_{0}^{1}{x}^{2} \cos (\frac{\pi }{2}{x}^{3})dx.[/latex]
Substitution may be only one of the techniques needed to evaluate a definite integral. All of the properties and rules of integration apply independently, and trigonometric functions may need to be rewritten using a trigonometric identity before we can apply substitution. Also, we have the option of replacing the original expression for [latex]u[/latex] after we find the antiderivative, which means that we do not have to change the limits of integration. These two approaches are shown in (Figure).
Using Substitution to Evaluate a Trigonometric Integral
Use substitution to evaluate [latex]{\int }_{0}^{\pi \text{/}2}{ \cos }^{2}\theta d\theta .[/latex]
Evaluating a Definite Integral Involving an Exponential Function
Evaluate the definite integral [latex]{\int }_{1}^{2}{e}^{1-x}dx.[/latex]
![A graph of the function f(x) = e^(1-x) over [0, 3]. It crosses the y axis at (0, e) as a decreasing concave up curve and symptotically approaches 0 as x goes to infinity.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/2332/2018/01/11204254/CNX_Calc_Figure_05_06_002.jpg)
Evaluate [latex]{\int }_{0}^{2}{e}^{2x}dx.[/latex]
Evaluating a Definite Integral Using Substitution
Evaluate the definite integral using substitution: [latex]{\int }_{1}^{2}\frac{{e}^{1\text{/}x}}{{x}^{2}}dx.[/latex]
Evaluate the definite integral using substitution: [latex]{\int }_{1}^{2}\frac{1}{{x}^{3}}{e}^{4{x}^{-2}}dx.[/latex]
Hint
Let [latex]u=4{x}^{-2}.[/latex]
(Figure) is a definite integral of a trigonometric function. With trigonometric functions, we often have to apply a trigonometric property or an identity before we can move forward. Finding the right form of the integrand is usually the key to a smooth integration.
Evaluating a Definite Integral
Find the definite integral of [latex]{\int }_{0}^{\pi \text{/}2}\frac{ \sin x}{1+ \cos x}dx.[/latex]
Evaluating a Definite Integral Using Inverse Trigonometric Functions
Evaluate the definite integral [latex]{\int }_{0}^{1}\frac{dx}{\sqrt{1-{x}^{2}}}.[/latex]
Evaluating a Definite Integral
Evaluate the definite integral [latex]{\int }_{0}^{\sqrt{3}\text{/}2}\frac{du}{\sqrt{1-{u}^{2}}}.[/latex]
Evaluating a Definite Integral
Evaluate the definite integral [latex]{\int }_{\sqrt{3}\text{/}3}^{\sqrt{3}}\frac{dx}{1+{x}^{2}}.[/latex]
Evaluate the definite integral [latex]{\int }_{0}^{2}\frac{dx}{4+{x}^{2}}.[/latex]
Hint
Follow the procedures from (Figure) to solve the problem.
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.
1. [T][latex]y=3{(1-x)}^{2}[/latex] over [latex]\left[0,2\right][/latex]
2. [T][latex]y=x{(1-{x}^{2})}^{3}[/latex] over [latex]\left[-1,2\right][/latex]
3. [T][latex]y= \sin x{(1- \cos x)}^{2}[/latex] over [latex]\left[0,\pi \right][/latex]
4. [T][latex]y=\frac{x}{{({x}^{2}+1)}^{2}}[/latex] over [latex]\left[-1,1\right][/latex]
In the following exercises, use a change of variables to evaluate the definite integral.
5. [latex]{\int }_{0}^{1}x\sqrt{1-{x}^{2}}dx[/latex]
6. [latex]{\int }_{0}^{1}\frac{x}{\sqrt{1+{x}^{2}}}dx[/latex]
7. [latex]{\int }_{0}^{2}\frac{t}{\sqrt{5+{t}^{2}}}dt[/latex]
8. [latex]{\int }_{0}^{1}\frac{t}{\sqrt{1+{t}^{3}}}dt[/latex]
9. [latex]{\int }_{0}^{\pi \text{/}4}{ \sec }^{2}\theta \tan \theta d\theta [/latex]
10. [latex]{\int }_{0}^{\pi \text{/}4}\frac{ \sin \theta }{{ \cos }^{4}\theta }d\theta [/latex]
In the following exercises, evaluate the indefinite integral [latex]\int f(x)dx[/latex] with constant [latex]C=0[/latex] using [latex]u[/latex]-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 [latex]F(x)={\int }_{a}^{x}f(t)dt,[/latex] with [latex]a[/latex] the left endpoint of the given interval.
11. [T][latex]\int (2x+1){e}^{{x}^{2}+x-6}dx[/latex] over [latex]\left[-3,2\right][/latex]
12. [T][latex]\int \frac{ \cos (\text{ln}(2x))}{x}dx[/latex] on [latex]\left[0,2\right][/latex]
13. [T][latex]\int \frac{3{x}^{2}+2x+1}{\sqrt{{x}^{3}+{x}^{2}+x+4}}dx[/latex] over [latex]\left[-1,2\right][/latex]
14. [T][latex]\int \frac{ \sin x}{{ \cos }^{3}x}dx[/latex] over [latex]\left[-\frac{\pi }{3},\frac{\pi }{3}\right][/latex]
![Two graphs. The first is the function f(x) = sin(x) / cos(x)^3 over [-5pi/16, 5pi/16]. It is an increasing concave down function for values less than zero and an increasing concave up function for values greater than zero. The second is the fuction f(x) = ½ sec(x)^2 over the same interval. It is a wide, concave up curve which decreases for values less than zero and increases for values greater than zero.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/2332/2018/01/11204239/CNX_Calc_Figure_05_05_206.jpg)
15. [T][latex]\int (x+2){e}^{\text{−}{x}^{2}-4x+3}dx[/latex] over [latex]\left[-5,1\right][/latex]
16. [T][latex]\int 3{x}^{2}\sqrt{2{x}^{3}+1}dx[/latex] over [latex]\left[0,1\right][/latex]
17. If [latex]h(a)=h(b)[/latex] in [latex]{\int }_{a}^{b}g\text{‘}(h(x))h(x)dx,[/latex] what can you say about the value of the integral?
18. Is the substitution [latex]u=1-{x}^{2}[/latex] in the definite integral [latex]{\int }_{0}^{2}\frac{x}{1-{x}^{2}}dx[/latex] okay? If not, why not?
In the following exercises, use a change of variables to show that each definite integral is equal to zero.
19. [latex]{\int }_{0}^{\pi }{ \cos }^{2}(2\theta ) \sin (2\theta )d\theta [/latex]
20. [latex]{\int }_{0}^{\sqrt{\pi }}t \cos ({t}^{2}) \sin ({t}^{2})dt[/latex]
21. [latex]{\int }_{0}^{1}(1-2t)dt[/latex]
22. [latex]{\int }_{0}^{1}\frac{1-2t}{(1+{(t-\frac{1}{2})}^{2})}dt[/latex]
23. [latex]{\int }_{0}^{\pi } \sin ({(t-\frac{\pi }{2})}^{3}) \cos (t-\frac{\pi }{2})dt[/latex]
24. [latex]{\int }_{0}^{2}(1-t) \cos (\pi t)dt[/latex]
25. [latex]{\int }_{\pi \text{/}4}^{3\pi \text{/}4}{ \sin }^{2}t \cos tdt[/latex]
26. Show that the average value of [latex]f(x)[/latex] over an interval [latex]\left[a,b\right][/latex] is the same as the average value of [latex]f(cx)[/latex] over the interval [latex]\left[\frac{a}{c},\frac{b}{c}\right][/latex] for [latex]c>0.[/latex]
27. Find the area under the graph of [latex]f(t)=\frac{t}{{(1+{t}^{2})}^{a}}[/latex] between [latex]t=0[/latex] and [latex]t=x[/latex] where [latex]a>0[/latex] and [latex]a\ne 1[/latex] is fixed, and evaluate the limit as [latex]x\to \infty .[/latex]
28. Find the area under the graph of [latex]g(t)=\frac{t}{{(1-{t}^{2})}^{a}}[/latex] between [latex]t=0[/latex] and [latex]t=x,[/latex] where [latex]0<x<1[/latex] and [latex]a>0[/latex] is fixed. Evaluate the limit as [latex]x\to 1.[/latex]
29. The area of a semicircle of radius 1 can be expressed as [latex]{\int }_{-1}^{1}\sqrt{1-{x}^{2}}dx.[/latex] Use the substitution [latex]x= \cos t[/latex] to express the area of a semicircle as the integral of a trigonometric function. You do not need to compute the integral.
30. The area of the top half of an ellipse with a major axis that is the [latex]x[/latex]-axis from [latex]x=-1[/latex] to [latex]a[/latex] and with a minor axis that is the [latex]y[/latex]-axis from [latex]y=\text{−}b[/latex] to [latex]b[/latex] can be written as [latex]{\int }_{\text{−}a}^{a}b\sqrt{1-\frac{{x}^{2}}{{a}^{2}}}dx.[/latex] Use the substitution [latex]x=a \cos t[/latex] to express this area in terms of an integral of a trigonometric function. You do not need to compute the integral.
31. [T] The following graph is of a function of the form [latex]f(t)=a \sin (nt)+b \sin (mt).[/latex] Estimate the coefficients [latex]a[/latex] and [latex]b[/latex], and the frequency parameters [latex]n[/latex] and [latex]m[/latex]. Use these estimates to approximate [latex]{\int }_{0}^{\pi }f(t)dt.[/latex]
![A graph of a function of the given form over [0, 2pi], which has six turning points. They are located at just before pi/4, just after pi/2, between 3pi/4 and pi, between pi and 5pi/4, just before 3pi/2, and just after 7pi/4 at about 3, -2, 1, -1, 2, and -3. It begins at the origin and ends at (2pi, 0). It crosses the x axis between pi/4 and pi/2, just before 3pi/4, pi, just after 5pi/4, and between 3pi/2 and 4pi/4.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/2332/2018/01/11204247/CNX_Calc_Figure_05_05_201.jpg)
32. [T] The following graph is of a function of the form [latex]f(x)=a \cos (nt)+b \cos (mt).[/latex] Estimate the coefficients [latex]a[/latex] and [latex]b[/latex] and the frequency parameters [latex]n[/latex] and [latex]m[/latex]. Use these estimates to approximate [latex]{\int }_{0}^{\pi }f(t)dt.[/latex]
![The graph of a function of the given form over [0, 2pi]. It begins at (0,1) and ends at (2pi, 1). It has five turning points, located just after pi/4, between pi/2 and 3pi/4, pi, between 5pi/4 and 3pi/2, and just before 7pi/4 at about -1.5, 2.5, -3, 2.5, and -1. It crosses the x axis between 0 and pi/4, just before pi/2, just after 3pi/4, just before 5pi/4, just after 3pi/2, and between 7pi/4 and 2pi.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/2332/2018/01/11204249/CNX_Calc_Figure_05_05_202.jpg)
33. [latex]{\int }_{0}^{\pi \text{/}4} \tan xdx[/latex]
34. [latex]{\int }_{0}^{\pi \text{/}3}\frac{ \sin x- \cos x}{ \sin x+ \cos x}dx[/latex]
35. [latex]{\int }_{\pi \text{/}6}^{\pi \text{/}2} \csc xdx[/latex]
36. [latex]{\int }_{\pi \text{/}4}^{\pi \text{/}3} \cot xdx[/latex]
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.
37. [T][latex]y={e}^{x}[/latex] over [latex]\left[0,1\right][/latex]
38. [T][latex]y={e}^{\text{−}x}[/latex] over [latex]\left[0,1\right][/latex]
39. [T][latex]y=\text{ln}(x)[/latex] over [latex]\left[1,2\right][/latex]
40. [T][latex]y=\frac{x+1}{{x}^{2}+2x+6}[/latex] over [latex]\left[0,1\right][/latex]
41. [T][latex]y={2}^{x}[/latex] over [latex]\left[-1,0\right][/latex]
42. [T][latex]y=\text{−}{2}^{\text{−}x}[/latex] over [latex]\left[0,1\right][/latex]
In the following exercises, [latex]f(x)\ge 0[/latex] for [latex]a\le x\le b.[/latex] Find the area under the graph of [latex]f(x)[/latex] between the given values [latex]a[/latex] and [latex]b[/latex] by integrating.
43. [latex]f(x)=\frac{{\text{log}}_{10}(x)}{x};a=10,b=100[/latex]
44. [latex]f(x)=\frac{{\text{log}}_{2}(x)}{x};a=32,b=64[/latex]
45. [latex]f(x)={2}^{\text{−}x};a=1,b=2[/latex]
46. [latex]f(x)={2}^{\text{−}x};a=3,b=4[/latex]
47. Find the area under the graph of the function [latex]f(x)=x{e}^{\text{−}{x}^{2}}[/latex] between [latex]x=0[/latex] and [latex]x=5.[/latex]
48. Compute the integral of [latex]f(x)=x{e}^{\text{−}{x}^{2}}[/latex] and find the smallest value of N such that the area under the graph [latex]f(x)=x{e}^{\text{−}{x}^{2}}[/latex] between [latex]x=N[/latex] and [latex]x=N+10[/latex] is, at most, 0.01.
49. Find the limit, as N tends to infinity, of the area under the graph of [latex]f(x)=x{e}^{\text{−}{x}^{2}}[/latex] between [latex]x=0[/latex] and [latex]x=5.[/latex]
50. Show that [latex]{\int }_{a}^{b}\frac{dt}{t}={\int }_{1\text{/}b}^{1\text{/}a}\frac{dt}{t}[/latex] when [latex]0<a\le b.[/latex]
51. Suppose that [latex]f(x)>0[/latex] for all [latex]x[/latex] and that [latex]f[/latex] and [latex]g[/latex] are differentiable. Use the identity [latex]{f}^{g}={e}^{g\text{ln}f}[/latex] and the chain rule to find the derivative of [latex]{f}^{g}.[/latex]
52. Use the previous exercise to find the antiderivative of [latex]h(x)={x}^{x}(1+\text{ln}x)[/latex] and evaluate [latex]{\int }_{2}^{3}{x}^{x}(1+\text{ln}x)dx.[/latex]
53. Show that if [latex]c>0,[/latex] then the integral of [latex]1\text{/}x[/latex] from ac to bc [latex](0<a<b)[/latex] is the same as the integral of [latex]1\text{/}x[/latex] from [latex]a[/latex] to [latex]b[/latex].
The following exercises are intended to derive the fundamental properties of the natural log starting from the Definition/em> [latex]\text{ln}(x)={\int }_{1}^{x}\frac{dt}{t},[/latex] using properties of the definite integral and making no further assumptions.
54. Use the identity [latex]\text{ln}(x)={\int }_{1}^{x}\frac{dt}{t}[/latex] to derive the identity [latex]\text{ln}(\frac{1}{x})=\text{−}\text{ln}x.[/latex]
55. Use a change of variable in the integral [latex]{\int }_{1}^{xy}\frac{1}{t}dt[/latex] to show that [latex]\text{ln}xy=\text{ln}x+\text{ln}y\text{ for }x,y>0.[/latex]
56. Use the identity [latex]\text{ln}x={\int }_{1}^{x}\frac{dt}{x}[/latex] to show that [latex]\text{ln}(x)[/latex] is an increasing function of [latex]x[/latex] on [latex][0,\infty )[/latex] and use the previous exercises to show that the range of [latex]\text{ln}(x)[/latex] is [latex](\text{−}\infty ,\infty ).[/latex] Without any further assumptions, conclude that [latex]\text{ln}(x)[/latex] has an inverse function defined on [latex](\text{−}\infty ,\infty ).[/latex]
57. Pretend, for the moment, that we do not know that [latex]{e}^{x}[/latex] is the inverse function of [latex]\text{ln}(x),[/latex] but keep in mind that [latex]\text{ln}(x)[/latex] has an inverse function defined on [latex](\text{−}\infty ,\infty ).[/latex] Call it E. Use the identity [latex]\text{ln}xy=\text{ln}x+\text{ln}y[/latex] to deduce that [latex]E(a+b)=E(a)E(b)[/latex] for any real numbers [latex]a[/latex], [latex]b[/latex].
58. Pretend, for the moment, that we do not know that [latex]{e}^{x}[/latex] is the inverse function of [latex]\text{ln}x,[/latex] but keep in mind that [latex]\text{ln}x[/latex] has an inverse function defined on [latex](\text{−}\infty ,\infty ).[/latex] Call it E. Show that [latex]E\text{‘}(t)=E(t).[/latex]
59. The sine integral, defined as [latex]S(x)={\int }_{0}^{x}\frac{ \sin t}{t}dt[/latex] is an important quantity in engineering. Although it does not have a simple closed formula, it is possible to estimate its behavior for large [latex]x[/latex]. Show that for [latex]k\ge 1,|S(2\pi k)-S(2\pi (k+1))|\le \frac{1}{k(2k+1)\pi }.[/latex](Hint: [latex]\sin (t+\pi )=\text{−} \sin t[/latex])
60. [T] The normal distribution in probability is given by [latex]p(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{\text{−}{(x-\mu )}^{2}\text{/}2{\sigma }^{2}},[/latex] where σ is the standard deviation and μ is the average. The standard normal distribution in probability, [latex]{p}_{s},[/latex] corresponds to [latex]\mu =0\text{ and }\sigma =1.[/latex] Compute the left endpoint estimates [latex]{R}_{10}\text{ and }{R}_{100}[/latex] of [latex]{\int }_{-1}^{1}\frac{1}{\sqrt{2\pi }}{e}^{\text{−}{x}^{2\text{/}2}}dx.[/latex]
61. [T] Compute the right endpoint estimates [latex]{R}_{50}\text{ and }{R}_{100}[/latex] of [latex]{\int }_{-3}^{5}\frac{1}{2\sqrt{2\pi }}{e}^{\text{−}{(x-1)}^{2}\text{/}8}.[/latex]
In the following exercises, evaluate each integral in terms of an inverse trigonometric function.
62. [latex]{\int }_{0}^{\sqrt{3}\text{/}2}\frac{dx}{\sqrt{1-{x}^{2}}}[/latex]
63. [latex]{\int }_{-1\text{/}2}^{1\text{/}2}\frac{dx}{\sqrt{1-{x}^{2}}}[/latex]
64. [latex]{\int }_{\sqrt{3}}^{1}\frac{dx}{\sqrt{1+{x}^{2}}}[/latex]
65. [latex]{\int }_{1\text{/}\sqrt{3}}^{\sqrt{3}}\frac{dx}{1+{x}^{2}}[/latex]
66. [latex]{\int }_{1}^{\sqrt{2}}\frac{dx}{|x|\sqrt{{x}^{2}-1}}[/latex]
67. [latex]{\int }_{1}^{2\text{/}\sqrt{3}}\frac{dx}{|x|\sqrt{{x}^{2}-1}}[/latex]
68. Explain what is wrong with the following integral: [latex]{\int }_{1}^{2}\frac{dt}{\sqrt{1-{t}^{2}}}.[/latex]
69. Explain what is wrong with the following integral: [latex]{\int }_{-1}^{1}\frac{dt}{|t|\sqrt{{t}^{2}-1}}.[/latex]
In the following exercises, solve for the antiderivative [latex]\int f[/latex] of [latex]f[/latex] with [latex]C=0,[/latex] then use a calculator to graph [latex]f[/latex] and the antiderivative over the given interval [latex]\left[a,b\right].[/latex] Identify a value of C such that adding C to the antiderivative recovers the definite integral [latex]F(x)={\int }_{a}^{x}f(t)dt.[/latex]
70. [T][latex]\int \frac{1}{\sqrt{9-{x}^{2}}}dx[/latex] over [latex]\left[-3,3\right][/latex]
71. [T][latex]\int \frac{9}{9+{x}^{2}}dx[/latex] over [latex]\left[-6,6\right][/latex]
72. [T][latex]\int \frac{ \cos x}{4+{ \sin }^{2}x}dx[/latex] over [latex]\left[-6,6\right][/latex]
![Two graphs. The first shows the function f(x) = cos(x) / (4 + sin(x)^2). It is an oscillating function over [-6, 6] with turning points at roughly (-3, -2.5), (0, .25), and (3, -2.5), where (0,.25) is a local max and the others are local mins. The second shows the function F(x) = .5 * arctan(.5*sin(x)), which also oscillates over [-6,6]. It has turning points at roughly (-4.5, .25), (-1.5, -.25), (1.5, .25), and (4.5, -.25).](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/2332/2018/01/11204305/CNX_Calc_Figure_05_07_203.jpg)
73. [T][latex]\int \frac{{e}^{x}}{1+{e}^{2x}}dx[/latex] over [latex]\left[-6,6\right][/latex]
In the following exercises, use a calculator to graph the antiderivative [latex]\int f[/latex] with [latex]C=0[/latex] over the given interval [latex]\left[a,b\right].[/latex] Approximate a value of C, if possible, such that adding C to the antiderivative gives the same value as the definite integral [latex]F(x)={\int }_{a}^{x}f(t)dt.[/latex]
74. [T][latex]\int \frac{1}{x\sqrt{{x}^{2}-4}}dx[/latex] over [latex]\left[2,6\right][/latex]
75. [T][latex]\int \frac{1}{(2x+2)\sqrt{x}}dx[/latex] over [latex]\left[0,6\right][/latex]
76. [T][latex]\int \frac{( \sin x+x \cos x)}{1+{x}^{2}{ \sin }^{2}x}dx[/latex] over [latex]\left[-6,6\right][/latex]
![The graph of f(x) = arctan(x sin(x)) over [-6,6]. It has five turning points at roughly (-5, -1.5), (-2,1), (0,0), (2,1), and (5,-1.5).](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/2332/2018/01/11204310/CNX_Calc_Figure_05_07_207.jpg)
The general antiderivative is [latex]{ \tan }^{-1}(x \sin x)+C.[/latex] Taking [latex]C=\text{−}{ \tan }^{-1}(6 \sin (6))[/latex] recovers the definite integral.
77. [T][latex]\int \frac{2{e}^{-2x}}{\sqrt{1-{e}^{-4x}}}dx[/latex] over [latex]\left[0,2\right][/latex]
78. [T][latex]\int \frac{1}{x+x{\text{ln}}^{2}x}[/latex] over [latex]\left[0,2\right][/latex]
![A graph of the function f(x) = arctan(ln(x)) over (0, 2]. It is an increasing curve with x-intercept at (1,0).](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/2332/2018/01/11204313/CNX_Calc_Figure_05_07_209.jpg)
The general antiderivative is [latex]{ \tan }^{-1}(\text{ln}x)+C.[/latex] Taking [latex]C=\frac{\pi }{2}={ \tan }^{-1}\infty [/latex] recovers the definite integral.
79. [T][latex]\int \frac{{ \sin }^{-1}x}{\sqrt{1-{x}^{2}}}[/latex] over [latex]\left[-1,1\right][/latex]
In the following exercises, compute each definite integral.
80. [latex]{\int }_{0}^{1\text{/}2}\frac{ \tan ({ \sin }^{-1}t)}{\sqrt{1-{t}^{2}}}dt[/latex]
81. [latex]{\int }_{1\text{/}4}^{1\text{/}2}\frac{ \tan ({ \cos }^{-1}t)}{\sqrt{1-{t}^{2}}}dt[/latex]
82. [latex]{\int }_{0}^{1\text{/}2}\frac{ \sin ({ \tan }^{-1}t)}{1+{t}^{2}}dt[/latex]
83. [latex]{\int }_{0}^{1\text{/}2}\frac{ \cos ({ \tan }^{-1}t)}{1+{t}^{2}}dt[/latex]
84. For [latex]A>0,[/latex] compute [latex]I(A)={\int }_{\text{−}A}^{A}\frac{dt}{1+{t}^{2}}[/latex] and evaluate [latex]\underset{a\to \infty }{\text{lim}}I(A),[/latex] the area under the graph of [latex]\frac{1}{1+{t}^{2}}[/latex] on [latex]\left[\text{−}\infty ,\infty \right].[/latex]
85. Use the following graph to prove that [latex]{\int }_{0}^{x}\sqrt{1-{t}^{2}}dt=\frac{1}{2}x\sqrt{1-{x}^{2}}+\frac{1}{2}\phantom{\rule{0.05em}{0ex}}{ \sin }^{-1}x.[/latex]
Glossary
- change of variables
- the substitution of a variable, such as [latex]u[/latex], for an expression in the integrand
- integration by substitution
- a technique for integration that allows integration of functions that are the result of a chain-rule derivative
Hint
Let [latex]u=2x.[/latex]