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. For $x>1$:  $\displaystyle\int \frac{{x}^{2}}{\sqrt{x-1}}dx = \frac{2}{15}\sqrt{x-1}(3{x}^{2}+4x+8)+C; \,\,\, u=x-1$
4. $\displaystyle\int \frac{x}{\sqrt{4{x}^{2}+9}}dx=\frac{1}{4}\sqrt{4{x}^{2}+9}+C;\,\,\, u=4{x}^{2}+9$
5. $\displaystyle\int {(x+1)}^{4}dx; \,\,\, u=x+1$
6. $\displaystyle\int {(2x-3)}^{-7}dx; \,\,\, u=2x-3$
7. $\displaystyle\int \frac{x}{\sqrt{{x}^{2}+1}}dx; \,\,\, u={x}^{2}+1$
8. $\displaystyle\int (x-1){({x}^{2}-2x)}^{3}dx; \,\,\, u={x}^{2}-2x$
9. $\displaystyle\int { \cos }^{3}\theta d\theta ; \,\,\, u= \sin \theta$
   Hint

   ($\cos ^{2}\theta =1-{ \sin }^{2}\theta$)

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

10. $\displaystyle\int x{(1-x)}^{99}dx$
11. $\displaystyle\int {(11x-7)}^{-3}dx$
12. $\displaystyle\int { \cos }^{3}\theta \sin \theta d\theta$
13. $\displaystyle\int { \cos }^{2}(\pi t) \sin (\pi t)dt$
14. $\displaystyle\int t \sin ({t}^{2}) \cos ({t}^{2})dt$
15. $\displaystyle\int \frac{{x}^{2}}{{({x}^{3}-3)}^{2}}dx$
16. $\displaystyle\int \frac{{y}^{5}}{{(1-{y}^{3})}^{3\text{/}2}}dy$
17. ${\displaystyle\int (1-{ \cos }^{3}\theta )}^{10}{ \cos }^{2}\theta \sin \theta d\theta$
18. $\displaystyle\int ({ \sin }^{2}\theta -2 \sin \theta ){({ \sin }^{3}\theta -3{ \sin }^{2}\theta )}^{3} \cos \theta d\theta$

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.

19. $y=x{(1-{x}^{2})}^{3}$ over $\left[-1,2\right]$
20. $y=\dfrac{x}{{({x}^{2}+1)}^{2}}$ over $\left[-1,1\right]$

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

21. ${\displaystyle\int }_{0}^{1}\dfrac{x}{\sqrt{1+{x}^{2}}}dx$
22. ${\displaystyle\int }_{0}^{1}\dfrac{t^2}{\sqrt{1+{t}^{3}}}dt$
23. ${\displaystyle\int }_{0}^{\pi \text{/}4}\dfrac{ \sin \theta }{{ \cos }^{4}\theta }d\theta$

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.

24. $\displaystyle\int \frac{ \cos (\text{ln}(2x))}{x}dx$ on $\left[0,2\right]$
25. $\displaystyle\int \frac{ \sin x}{{ \cos }^{3}x}dx$ over $\left[-\frac{\pi }{3},\frac{\pi }{3}\right]$
26. $\displaystyle\int 3{x}^{2}\sqrt{2{x}^{3}+1}dx$ over $\left[0,1\right]$
27. 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.

28. ${\displaystyle\int }_{0}^{\sqrt{\pi }}t \cos ({t}^{2}) \sin ({t}^{2})dt$
29. ${\displaystyle\int }_{0}^{1}\dfrac{1-2t}{(1+{(t-\frac{1}{2})}^{2})}dt$
30. ${\displaystyle\int }_{0}^{2}(1-t) \cos (\pi t)dt$
31. 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 $c>0.$
32. Find the area under the graph of $g(t)=\dfrac{t}{{(1-{t}^{2})}^{a}}$ between $t=0$ and $t=x,$ where [latex]00[/latex] is fixed. Evaluate the limit as $x\to 1.$
33. 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.
34. 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 ${\displaystyle\int }_{0}^{\pi }f(t)dt.$

    

Integrals Involving Exponential and Logarithmic Functions

In the following exercises, verify by differentiation that $\displaystyle\int \text{ln}xdx=x(\text{ln}x-1)+C,$ then use appropriate changes of variables to compute the integral.

1. $\displaystyle\int {x}^{2}{\text{ln}}^{2}xdx$
2. $\displaystyle\int \frac{\text{ln}x}{\sqrt{x}}dx$$(\text{Hint: }\text{ Set }u=\sqrt{x}\text{.})$
3. Write an integral to express the area under the graph of $y={e}^{t}$ between $t=0$ and $t=\text{ln}x,$ and evaluate the integral.

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

4. $\displaystyle\int \frac{ \sin (3x)- \cos (3x)}{ \sin (3x)+ \cos (3x)}dx$
5. $\displaystyle\int x \csc ({x}^{2})dx$
6. $\displaystyle\int \text{ln}( \csc x) \cot xdx$

In the following exercise, evaluate the definite integral.

7. ${\displaystyle\int }_{1}^{2}\frac{1+2x+{x}^{2}}{3x+3{x}^{2}+{x}^{3}}dx$

In the following exercises, integrate using the indicated substitution.

8. $\displaystyle\int \frac{y-1}{y+1}dy;u=y+1$
9. $\displaystyle\int \frac{ \sin x+ \cos x}{ \sin x- \cos x}dx;u= \sin x- \cos x$
10. $\displaystyle\int \text{ln}(x)\frac{\sqrt{1-{(\text{ln}x)}^{2}}}{x}dx;u=\text{ln}x$

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.

11. $y={e}^{\text{−}x}$ over $\left[0,1\right]$
12. $y=\frac{x+1}{{x}^{2}+2x+6}$ over $\left[0,1\right]$
13. $y=\text{−}{2}^{\text{−}x}$ over $\left[0,1\right]$

Integrals Resulting in Inverse Trigonometric

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

1. ${\displaystyle\int }_{0}^{\sqrt{3}\text{/}2}\frac{dx}{\sqrt{1-{x}^{2}}}$

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