Techniques of Integration

What it means for an improper integral to converge or to diverge?

Use limits of definite integrals to calculate each of the improper integrals in Exercises 21–56. 

${\int }_0^{\infty }\frac{x}{\left(x^2+1\right)}dx$

Solve the integral

$\int \sqrt{3-x}\sqrt{x-1}dx$

Solve each of the integrals in Exercises 21–66. Some of the integrals require the methods presented in this section, and some do not. (The last four exercises involve hyperbolic functions.) 

$\int {\cos }^3\left(x\right){\sec }^2\left(x\right)dx$

Solve each of the integrals in Exercises $75–78$ by using polynomial long division to rewrite the integrand. This is one way that you can sometimes avoid using trigonometric substitution; moreover, sometimes it works when trigonometric substitution does not apply.

$\int \frac{x^3}{x^2+4}dx$

Read the section and make your own summary of the material.

Problem Zero: Read the section and make your own summary of the material. 

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) \(\int g'(h(x))h'(x)dx=g(h(x))+C\)

(b) If \(v=u^2+1\), then \(\int \sqrt{u^2+1}du=\int \sqrt{v}dv\)

(c) If \(u=x^3\), then \(\int x\sin x^3dx=\frac{1}{3x}\int \sin udu\)

(d) \(\int_0^3 u^2du=\int_0^3(u(x))^3du\)

(e) \(\int_0^1x^2dx=\int_0^1u^2du\)

(f) \(\int_2^4xe^{x^2-1}dx=\frac{1}{2}\int_2^4e^udu\)

(g) \(\int_2^3f(u(x))u'(x)dx=\int_{u(2)}^{u(3)}f(u)du\)

(h) \(\int_0^6f(u(x))u'(x)dx=\left[\int f(u)du\right]_0^6\)

Expressing geometric quantities with integrals: Express each of the given geometric quantities in terms of definite integrals. You do not have to solve the integrals. 

The signed area of the region between the graph of $f\left(x\right)=\sin x$ and the x-axis on $\left[0,\frac{3\pi }{2}\right]$

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample. 

(a) True/False: $\int g^{\mathrm{\prime }}\left(h\right(x\left)\right)h^{\mathrm{\prime }}\left(x\right)dx=g\left(h\right(x\left)\right)+C$

(b) True/False: If $v=u^2+1,$ then $\int \sqrt{u^2+1}du=\int \sqrt{v} dv$

(c) True/False: If $u=x^3,$ then $\int x\sin \left(x^3\right)dx=\frac{1}{3x}\int \sin u du$

(d) True/False: ${\int }_0^3 u^{2}du={\int }_{x=0}^{x=3} \left(u\right(x){)}^{2}du$

(e) True/False: ${\int }_0^1 x^{2}dx={\int }_0^1 u^{2}du$

(f) True/False: ${\int }_2^4 x{e}^{x^2-1}dx=\frac{1}{2}{\int }_2^4 e^{u}du$ 

(g) True/False: ${\int }_2^3 f\left(u\right(x\left)\right)u^{\mathrm{\prime }}\left(x\right)dx={\int }_{u\left(2\right)}^{u\left(3\right)} f\left(u\right)du$

(h) True/False: 

Construct examples of the thing(s) described in the following.

(a) Five integrals that can be solved with the method of

integration by substitution.

(b) Five integrals that cannot be solved with the method

of integration by substitution.

(c) Three relatively simple integrals that we cannot solve

with any of the methods we now know.

Explain why \(\int \frac{2x}{x^2+1}dx\) and \(\int \frac{1}{x\ln x}dx\)are essentially the same integral after a change of variables.

List some things which would suggest that a certain substitution \(u(x)\) could be a useful choice. What do you look for when choosing \(u(x)?\)

Explain why $\int \frac{2x}{x^2+1}dx$ and $\int \frac{1}{x\ln x}dx$ are essentially the same integral after a change of variables.

List some things which would suggest that a certain substitution u(x) could be a useful choice. What do you look for when choosing u(x)?

For each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.

$\int \sin udu$

For each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.

$\int u^{2}du$

For each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.

$\int \frac{1}{\sqrt{u}}du$

For each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.

$\int e^{u}du$

For each function u(x) in Exercises 9–12, write the differential du in terms of the differential dx.

$u\left(x\right)=x^2+x+1$

For each function u(x) in Exercises 9–12, write the differential du in terms of the differential dx.

$u\left(x\right)=x^2+1$

For each function u(x) in Exercises 9–12, write the differential du in terms of the differential dx.

$u\left(x\right)=\sin x$

For each function u(x) in Exercises 9–12, write the differential du in terms of the differential dx.

$u\left(x\right)=\frac{1}{x}$

Suppose $u\left(x\right)=x^2$. Calculate and compare the values of the following definite integrals:

${\int }_{-1}^5{u}^{2}du,{\int }_{x=-1}^{x=5}{u}^{2}du \mathrm{and} {\int }_{u(-1)}^{u\left(5\right)}{u}^{2}du$

Find three integrals in Exercises 21–70 in which the denominator of the integrand is a good choice for a substitution u(x).

Find three integrals in Exercises 21–70 that we can anti-differentiate immediately after algebraic simplification.

Consider the integral $\int \sin x\cos x\mathrm{d}x$.

(a) Solve this integral by using u-substitution with $u=\sin x$ and $\mathrm{d}u=\cos x\mathrm{d}x$.

(b) Solve the integral another way, using u-substitution with $u=\cos x$ and $\mathrm{d}u=-\sin x\mathrm{d}x$.

(c) How must your two answers be related? Use algebra to prove this relationship.

Consider the integral $\int x{(x^2-1)}^2\mathrm{d}x$.

(a) Solve this integral by using u-substitution.

(b) Solve the integral another way, using algebra to multiply out the integrand first.

(c) How must your two answers be related? Use algebra to prove this relationship.

Consider the integral $\int \frac{x^{-2}-4}{x^3}dx$.

(a) Solve this integral by using u-substitution.

(b) Solve the integral another way, using algebra to simplify the integrand first.

(c) How must your two answers be related? Use algebra to prove this relationship.

Consider the integral ${\int }_{-2}^2{e}^{5-3x}dx$.

(a) Solve this integral by using u-substitution while keeping the limits of integration in terms of x.

(b) Solve the integral again with u-substitution, this time changing the  limits of integration to be in terms of u.

Consider the integral ${\int }_2^4\frac{x}{x^2-1}\mathrm{d}x$.

(a) Solve this integral by using u-substitution while keeping the limits of integration in terms of x.

(b) Solve the integral again with u-substitution, this time changing the  limits of integration to be in terms of u.

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

$\int {(3x+1)}^2\mathrm{d}x$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

$\int x{(x^2-1)}^2\mathrm{d}x$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

$\int \frac{8x}{x^2+1}\mathrm{d}x$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

$\int \left(\frac{2}{x-1}-\frac{3}{2x+1}\right)\mathrm{d}x$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

$\int \frac{x+3}{2\sqrt{x}}\mathrm{d}x$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.) 

$\int \frac{x^{2/3}+1}{\sqrt[3]{x}}\mathrm{d}x$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.) 

$\int x{\csc }^2{x}^2\mathrm{d}x$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.) 

$\int 3\cos \left(\pi x\right)\mathrm{d}x$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.) 

$\int \frac{1}{3x+1}\mathrm{d}x$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.) 

$\int x^3\cos x^4\mathrm{d}x$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.) 

$\int \sin \pi x\cos \pi x\mathrm{d}x$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.) 

$\int \frac{x}{\sqrt{3x^2+1}}dx$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.) 

$\int sec2x\tan 2x\mathrm{d}x$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.) 

$\int \frac{\sqrt{x}+1}{x}\mathrm{d}x$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.) 

$\int {\cot }^{5}x{\csc }^{2}x\mathrm{d}x$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.) 

$\int \sin x{e}^{\cos x}\mathrm{d}x$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.) 

$\int x^4{(x^3+1)}^2\mathrm{d}x$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.) 

$\int 2x{e}^{3x^2}\mathrm{d}x$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

$\int x^{1/4}\sin x^{5/4}\mathrm{d}x$