Substitution

1. Why is [latex]u[/latex]-substitution referred to as change of variable?
2. If [latex]f=g\circ h,[/latex] when reversing the chain rule, [latex]\frac{d}{dx}(g\circ h)(x)={g}^{\prime }(h(x)){h}^{\prime }(x),[/latex] should you take [latex]u=g(x)[/latex] or [latex]u=h(x)?[/latex]

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

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

   ([latex]\cos ^{2}\theta =1-{ \sin }^{2}\theta[/latex])

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

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

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

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

In the following exercises, evaluate the indefinite integral [latex]\displaystyle\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)={\displaystyle\int }_{a}^{x}f(t)dt,[/latex] with [latex]a[/latex] the left endpoint of the given interval.

24. [latex]\displaystyle\int \frac{ \cos (\text{ln}(2x))}{x}dx[/latex] on [latex]\left[0,2\right][/latex]
25. [latex]\displaystyle\int \frac{ \sin x}{{ \cos }^{3}x}dx[/latex] over [latex]\left[-\frac{\pi }{3},\frac{\pi }{3}\right][/latex]
26. [latex]\displaystyle\int 3{x}^{2}\sqrt{2{x}^{3}+1}dx[/latex] over [latex]\left[0,1\right][/latex]
27. Is the substitution [latex]u=1-{x}^{2}[/latex] in the definite integral [latex]{\displaystyle\int }_{0}^{2}\dfrac{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.

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

    

Integrals Involving Exponential and Logarithmic Functions

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

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

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

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

In the following exercise, evaluate the definite integral.

7. [latex]{\displaystyle\int }_{1}^{2}\frac{1+2x+{x}^{2}}{3x+3{x}^{2}+{x}^{3}}dx[/latex]

In the following exercises, integrate using the indicated substitution.

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

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

Integrals Resulting in Inverse Trigonometric

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

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

2. [latex]{\displaystyle\int }_{\sqrt{3}}^{1}\frac{dx}{\sqrt{1+{x}^{2}}}[/latex]
3. [latex]{\displaystyle\int }_{1}^{\sqrt{2}}\frac{dx}{|x|\sqrt{{x}^{2}-1}}[/latex]