## Problem Set: Substitution

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

2. If $f=g\circ h,$ when reversing the chain rule, $\frac{d}{dx}(g\circ h)(x)={g}^{\prime }(h(x)){h}^{\prime }(x),$ should you take $u=g(x)$ or $u=h(x)?$

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

3. $\displaystyle\int x\sqrt{x+1}dx=\frac{2}{15}{(x+1)}^{3\text{/}2}(3x-2)+C; \,\,\, u=x+1$

4. For $x>1$:  $\displaystyle\int \frac{{x}^{2}}{\sqrt{x-1}}dx = \frac{2}{15}\sqrt{x-1}(3{x}^{2}+4x+8)+C; \,\,\, u=x-1$

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

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

7. $\displaystyle\int \frac{x}{{(4{x}^{2}+9)}^{2}}dx=-\frac{1}{8(4{x}^{2}+9)};\,\,\, u=4{x}^{2}+9$

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

8. $\displaystyle\int {(x+1)}^{4}dx; \,\,\, u=x+1$

9. $\displaystyle\int {(x-1)}^{5}dx; \,\,\, u=x-1$

10. $\displaystyle\int {(2x-3)}^{-7}dx; \,\,\, u=2x-3$

11. $\displaystyle\int {(3x-2)}^{-11}dx; \,\,\, u=3x-2$

12. $\displaystyle\int \frac{x}{\sqrt{{x}^{2}+1}}dx; \,\,\, u={x}^{2}+1$

13. $\displaystyle\int \frac{x}{\sqrt{1-{x}^{2}}}dx; \,\,\, u=1-{x}^{2}$

14. $\displaystyle\int (x-1){({x}^{2}-2x)}^{3}dx; \,\,\, u={x}^{2}-2x$

15. $\displaystyle\int ({x}^{2}-2x){({x}^{3}-3{x}^{2})}^{2}dx; \,\,\, u={x}^{3}-3{x}^{2}$

16. $\displaystyle\int { \cos }^{3}\theta d\theta ; \,\,\, u= \sin \theta$

17. $\displaystyle\int { \sin }^{3}\theta d\theta ; \,\,\, u= \cos \theta$

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

18. $\displaystyle\int x{(1-x)}^{99}dx$

19. $\displaystyle\int t{(1-{t}^{2})}^{10}dt$

20. $\displaystyle\int {(11x-7)}^{-3}dx$

21. $\displaystyle\int {(7x-11)}^{4}dx$

22. $\displaystyle\int { \cos }^{3}\theta \sin \theta d\theta$

23. $\displaystyle\int { \sin }^{7}\theta \cos \theta d\theta$

24. $\displaystyle\int { \cos }^{2}(\pi t) \sin (\pi t)dt$

25. $\displaystyle\int { \sin }^{2}x{ \cos }^{3}xdx$

26. $\displaystyle\int $t$ \sin ({t}^{2}) \cos ({t}^{2})dt$

27. $\displaystyle\int {t}^{2}{ \cos }^{2}({t}^{3}) \sin ({t}^{3})dt$

28. $\displaystyle\int \frac{{x}^{2}}{{({x}^{3}-3)}^{2}}dx$

29. $\displaystyle\int \frac{{x}^{3}}{\sqrt{1-{x}^{2}}}dx$

30. $\displaystyle\int \frac{{y}^{5}}{{(1-{y}^{3})}^{3\text{/}2}}dy$

31. ${\displaystyle\int \cos \theta (1- \cos \theta )}^{99} \sin \theta d\theta$

32. ${\displaystyle\int (1-{ \cos }^{3}\theta )}^{10}{ \cos }^{2}\theta \sin \theta d\theta$

33. $\displaystyle\int ( \cos \theta -1){({ \cos }^{2}\theta -2 \cos \theta )}^{3} \sin \theta d\theta$

34. $\displaystyle\int ({ \sin }^{2}\theta -2 \sin \theta ){({ \sin }^{3}\theta -3{ \sin }^{2}\theta )}^{3} \cos \theta d\theta$

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] $y=3{(1-x)}^{2}$ over $\left[0,2\right]$

36. [T] $y=x{(1-{x}^{2})}^{3}$ over $\left[-1,2\right]$

37. [T] $y= \sin x{(1- \cos x)}^{2}$ over $\left[0,\pi \right]$

38. [T] $y=\dfrac{x}{{({x}^{2}+1)}^{2}}$ over $\left[-1,1\right]$

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

39. ${\displaystyle\int }_{0}^{1}x\sqrt{1-{x}^{2}}dx$

40. ${\displaystyle\int }_{0}^{1}\dfrac{x}{\sqrt{1+{x}^{2}}}dx$

41. ${\displaystyle\int }_{0}^{2}\dfrac{t^2}{\sqrt{5+{t}^{2}}}dt$

42. ${\displaystyle\int }_{0}^{1}\dfrac{t^2}{\sqrt{1+{t}^{3}}}dt$

43. ${\displaystyle\int }_{0}^{\pi \text{/}4}{ \sec }^{2}\theta \tan \theta d\theta$

44. ${\displaystyle\int }_{0}^{\pi \text{/}4}\dfrac{ \sin \theta }{{ \cos }^{4}\theta }d\theta$

In the following exercises, evaluate the indefinite integral $\displaystyle\int f(x)dx$ with constant $C=0$ 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 $F(x)={\displaystyle\int }_{a}^{x}f(t)dt,$ with $a$ the left endpoint of the given interval.

45. [T] $\displaystyle\int (2x+1){e}^{{x}^{2}+x-6}dx$ over $\left[-3,2\right]$

46. [T] $\displaystyle\int \frac{ \cos (\text{ln}(2x))}{x}dx$ on $\left[0,2\right]$

47. [T] $\displaystyle\int \frac{3{x}^{2}+2x+1}{\sqrt{{x}^{3}+{x}^{2}+x+4}}dx$ over $\left[-1,2\right]$

48. [T] $\displaystyle\int \frac{ \sin x}{{ \cos }^{3}x}dx$ over $\left[-\frac{\pi }{3},\frac{\pi }{3}\right]$

49. [T] $\displaystyle\int (x+2){e}^{\text{−}{x}^{2}-4x+3}dx$ over $\left[-5,1\right]$

50. [T] $\displaystyle\int 3{x}^{2}\sqrt{2{x}^{3}+1}dx$ over $\left[0,1\right]$

51. If $h(a)=h(b)$ in ${\displaystyle\int }_{a}^{b}g\text{‘}(h(x))h(x)dx,$ what can you say about the value of the integral?

52. Is the substitution $u=1-{x}^{2}$ in the definite integral ${\displaystyle\int }_{0}^{2}\dfrac{x}{1-{x}^{2}}dx$ okay? If not, why not?

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

53. ${\displaystyle\int }_{0}^{\pi }{ \cos }^{2}(2\theta ) \sin (2\theta )d\theta$

54. ${\displaystyle\int }_{0}^{\sqrt{\pi }}t \cos ({t}^{2}) \sin ({t}^{2})dt$

55. ${\displaystyle\int }_{0}^{1}(1-2t)dt$

56. ${\displaystyle\int }_{0}^{1}\dfrac{1-2t}{(1+{(t-\frac{1}{2})}^{2})}dt$

57. ${\displaystyle\int }_{0}^{\pi } \sin ({(t-\frac{\pi }{2})}^{3}) \cos (t-\frac{\pi }{2})dt$

58. ${\displaystyle\int }_{0}^{2}(1-t) \cos (\pi t)dt$

59. ${\displaystyle\int }_{\pi \text{/}4}^{3\pi \text{/}4}{ \sin }^{2}t \cos tdt$

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

61. Find the area under the graph of $f(t)=\dfrac{t}{{(1+{t}^{2})}^{a}}$ between $t=0$ and $t=x$ where $a>0$ and $a\ne 1$ is fixed, and evaluate the limit as $x\to \infty .$

62. Find the area under the graph of $g(t)=\dfrac{t}{{(1-{t}^{2})}^{a}}$ between $t=0$ and $t=x,$ where $0<x<1$ and $a>0$ is fixed. Evaluate the limit as $x\to 1.$

63. The area of a semicircle of radius 1 can be expressed as ${\displaystyle\int }_{-1}^{1}\sqrt{1-{x}^{2}}dx.$ Use the substitution $x= \cos t$ 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 $x=-1$ to $a$ and with a minor axis that is the $y$-axis from $y=\text{−}b$ to $b$ can be written as ${\displaystyle\int }_{\text{−}a}^{a}b\sqrt{1-\frac{{x}^{2}}{{a}^{2}}}dx.$ Use the substitution $x=a \cos t$ to express this area in terms of an integral of a trigonometric function. You do not need to compute the integral.

65. [T] The following graph is of a function of the form $f(t)=a \sin (nt)+b \sin (mt).$ Estimate the coefficients $a$ and $b$, and the frequency parameters $n$ and $m$. Use these estimates to approximate ${\displaystyle\int }_{0}^{\pi }f(t)dt.$

66. [T] The following graph is of a function of the form $f(x)=a \cos (nt)+b \cos (mt).$ Estimate the coefficients $a$ and $b$ and the frequency parameters $n$ and $m$. Use these estimates to approximate ${\displaystyle\int }_{0}^{\pi }f(t)dt.$