Techniques of Integration

Use the chain rule and the Fundamental Theorem of Calculus to prove the integration-by-substitution formula for definite integrals:

${\int }_a^{b}f\text{'}\left(u\right(x\left)\right)u\text{'}\left(x\right)dx=f\left(u\right(b\left)\right)-f\left(u\right(a\left)\right)$

Trigonometric integrals: The integrals that follow can be solved by using algebra to write the integrands in the form $f\text{'}\left(u\right(x\left)\right)u\text{'}\left(x\right)$ so that $u-$ substitution will apply.

Solve$\int {\sin }^{5}x\mathrm{d}x$ by using the Pythagorean identity ${\sin }^{2}x+{\cos }^{2}x=1$ to rewrite the integrand as ${(1-{\cos }^{2}x)}^2\sin x$ and then applying substitution with $u=\cos x$.

Trigonometric integrals: The integrals that follow can be solved by using algebra to write the integrands in the form $f\text{'}\left(u\right(x\left)\right)u\text{'}\left(x\right)$ so that $u-$ substitution will apply.

Solve $\int {\sin }^{3}x{\cos }^{4}x\mathrm{d}x$ by using the Pythagorean identity ${\sin }^{2}x+{\cos }^{2}x=1$ to rewrite the integrand as $(1-{\cos }^{2}x){\cos }^{4}x\sin x$ and then applying substitution with $u=\cos x$.

Trigonometric integrals: The integrals that follow can be solved by using algebra to write the integrands in the form $f\text{'}\left(u\right(x\left)\right)u\text{'}\left(x\right)$ so that$u-$ substitution will apply.

Solve $\int {\sec }^{4}x{\tan }^{3}x\mathrm{d}x$ by using the Pythagorean identity ${\tan }^{2}x+1={\sec }^{2}x$ to rewrite the integrand as $({\tan }^{2}x+1){\tan }^{3}x{\sec }^{2}x$ and then applying substitution with $u=\tan x$.

Prove the integration formula

$\int \tan x\mathrm{d}x=\ln \left|\sec x\right|+C$

(a) by using algebra and integration by substitution to find tan x dx;

(b) by differentiating ln | sec x|.

Prove the integration formula

$\int \cot x\mathrm{d}x=-\ln \left|\csc x\right|+C$

(a) by using algebra and integration by substitution to find $\int \cot x\mathrm{d}x$;

(b) by differentiating $-\ln \left|\csc x\right|$.

Differentiation review: Differentiate each of the functions that follow. Simplify your answers as much as possible.

a. $f\left(x\right)=x{e}^x-e^x$ 

b. $f\left(x\right)=\frac{x^2 \ln x}{2}-\frac{x^2}{4}$

c. $f\left(x\right)=-e^{-x}(x^2+1)-2x{e}^{-x}-2e^{-x}$ 

d. $f\left(x\right)=\frac{x^3}{3}- 2(x{e}^x-e^x ) + \frac{e^{2x}}{2}$

e. $f\left(x\right)=-x \cot x + \ln |\sin x|$

f. $f\left(x\right)=x^3 \sin x + 3x^2 \cos x - 6x \sin x - 6 \cos x$

Review of integration by substitution: Use u-substitution to find each of the following integrals.

a. $\int e^{3x+1} dx$ b. $\int x{e}^{x^2+1} dx$

c. $\int \frac{\ln x}{x} dx$ d. $\int \frac{1}{x \ln x} dx$

e. $\int \tan x {\sec }^{2}x dx$ f. $\int \frac{1}{x^2+1} dx$

g.$\int \sin x \sqrt{\cos x} dx$ h. $\int e^x \sin e^x dx$

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

True/Fälse: 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 or False: If $u=x^2+1$, then $d u=2 x$

(b) True or False: If $dv=x^{2}dx$, then $v=\frac{1}{3}{x}^3$

(c) True or False: We can apply integration by parts with $u=\ln x$ and $d v=x d x$ to the integral $\int \frac{\ln x}{x}dx$

(d) True or False: We can apply integration by parts with $u=x$ and $dv=\ln xdx$ to the integral $\int \frac{\ln x}{x}dx$

(e) True or False: Integration by parts has to do with reversing the product rule.

(f) True or False: Integration by parts is a good method for any integral that involves a product.

(g) True or False: In applying integration by parts, it is sometimes a good idea to choose $u$ to be the entire integrand and let $d v=d x$

(h) True or False: ${\int }_0^{3}x{e}^{x}dx=x{e}^x-{\int }_0^3{e}^{x}dx$

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) Five integrals for which u-substitution is a better strategy than integration by parts. List a good choice for u in each case.

(b) Five integrals for which integration by parts is a better strategy than u-substitution. List good choices for u and dv in each case.

(c) Three integrals that we cannot integrate with only the techniques we have learned so far.

State the integration-by-parts formula for indefinite integrals,

(a) using the notation $du$ and $dv$ and

(b) without using the notation $du$ and $dv$.

State the integration-by-parts formula for definite integrals,

(a) using the notation$du$ and $dv$ and

(b) without using the notation $du$and $dv$

Write down an integral that can be solved by using integration by parts with $u=x$ and another integral that can be solved by using integration by parts with $dv=x dx$

Write down an integral that can be solved by using integration by parts with $u=\sin x$ and another integral that can be solved by using integration by parts  with $dv=\sin x dx$.

Write down an integral that can be solved with integration by parts by choosing $u$ to be the entire integrand and $dv=dx$.

Suppose v(x) is a function of x. Explain why the integral

of dv is equal to v (up to a constant).

Explain why choosing $u=1$ (and thus choosing dv to be the entire integrand, including dx) is never a good choice for integration by parts. 

Find three integrals in Exercises 27–70 for which either algebra or u-substitution is a better strategy than integration by parts.

Find three integrals in Exercises 27–70 for which a good strategy is to use integration by parts with $u=x$ and dv the remaining part.

Find three integrals in Exercises 27–70 for which a good strategy is to apply integration by parts twice. 

If $u\left(x\right)=\sin 3x$ and $v\left(x\right)=x$, what are du and dv? Write down $\int udv$ and $\int vdu$ in this situation. Which of these integrals would be easier to find? What does this exercise have to do with integration by parts?

Provide a justification for each equality in the statement of the integration-by-parts formula for definite integrals from Theorem 5.10.

Explain why ${\left[g\left(x\right)\right]}_a^b-{\left[h\left(x\right)\right]}_a^b={\left[g\left(x\right)-h\left(x\right)\right]}_a^b$ this equation have to do with calculations of definite integrals with integration by parts?

For each pair of functions $u\left(x\right)$ and $v\left(x\right)$ in Exercises 16-18, fill in the blanks to complete each of the following:

(a) $\frac{d}{dx}\left(u\left(x\right)v\left(x\right)\right)$=______

(b) $\int \_\_\_\_dx=u\left(x\right)v\left(x\right)+C$

(c) $\int udv$=______

$u\left(x\right)=x$, $v\left(x\right)=\cos 2x$

For each pair of functions $u\left(x\right)$ and $v\left(x\right)$ in Exercises 16-18, fill in the blanks to complete each of the following:

(a) $\frac{d}{dx}\left(u\left(x\right)v\left(x\right)\right)$=_______

(b) $\int \_\_\_\_dx=u\left(x\right)v\left(x\right)+C$

(c) $\int udv=$______

$u\left(x\right)=\ln x$, $v\left(x\right)=x$

For each pair of functions u(x) and v(x) in Exercise, fill in the blanks to complete each of the following:

$u\left(x\right)=x^3,v\left(x\right)=e^{3x}$

(a) $\frac{d}{dx}\left(u\left(x\right)v\left(x\right)\right)=\_\_\_\_\_\_$

(b) width="178" style="max-width: none; vertical-align: -10px;" $\int \_\_\_\_ dx=u\left(x\right)v\left(x\right)+C$

(c) $\int u dv=\_\_\_\_\_$

Each expression that follows is the result of a calculation that uses integration by parts. That is, each is an expression of the form uv −  v du for some functions u and v. Identify u, v, du, and dv, and determine the original integral  u dv.

$\frac{x{2}^x}{\ln 2}-\frac{1}{\ln 2}\int 2^x dx$

Each expression that follows is the result of a calculation that uses integration by parts. That is, each is an expression of the form $uv - \int v du$ for some functions u and v. Identify u, v, du, and dv, and determine the original integral  u dv.

$-x^2 \cos x+2\int x \cos x dx$

Each expression that follows is the result of a calculation that uses integration by parts. That is, each is an expression of the form $uv - \int v du$ for some functions u and v. Identify u, v, du, and dv, and determine the original integral  u dv.

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

Each expression that follows is the result of a calculation that uses integration by parts. That is, each is an expression of the form $uv - \int v du$ for some functions u and v. Identify u, v, du, and dv, and determine the original integral $\int u dv$.

$\frac{1}{3}x{e}^{3x}-\frac{1}{3}\int e^{3x} dx$

Each expression that follows is the result of a calculation that uses integration by parts. That is, each is an expression of the form $uv - \int v du$ for some functions u and v. Identify u, v, du, and dv, and determine the original integral  $\int u dv$.

$x{\tan }^{-1}x-\int \frac{x}{x^2+1} dx$

Consider the integral $\int \frac{\ln x}{x} dx$.

(a) Solve this integral by using integration by parts with u = ln x and $dv=\frac{1}{x}dx$

(b) Now solve the integral another way, by using u-substitution with u = ln x.

(c) How must your answers to parts (a) and (b) be related? Use algebra to prove that this is so.

Solve the integral: $\int x \ln \sqrt{x} dx$

Solve the integral: $\int x^3{e}^{x^2}dx$

Solve the integral: $\int x^5 \cos x^{3}dx$

Consider the integral $\int x^2\ln x^3 dx$.

(a) Solve this integral by using integration by parts with $u = \ln \left(x^3\right)$ and $dv = x^2 dx$.

(b) Now solve the integral another way, by using u-substitution with u = x 3.

(c) How must your answers to parts (a) and (b) be related? Use algebra to prove this relationship.

Consider the integral $\int x{(x + 1)}^{100} dx$.

(a) Solve this integral by using integration by parts with u = x and $dv = {(x + 1)}^{100} dx$.

(b) Now solve the integral another way, by using u-substitution with u = x + 1 and back-substitution.

(c) How must your answers to parts (a) and (b) be related? Use graphs to prove this relationship.

Solve the integral: $\int x{e}^{x }dx$

Solve the integral: $\int x\sin x dx$

Solve the integral: $\int x\ln \left(x\right)dx$

Solve the integral: $\int x\cos \left(x\right)dx$.

Solve the integral:  $\int x\sin {\left(x\right)}^{2}dx$.

Solve the integral: $\int x^2\cos x dx$.

Solve the integral: $\int x^2{e}^{3x}dx$.

Solve the integral: $\int \left(3-x\right)e^{2x}dx$

Solve the integral: $\int \frac{x}{e^x}dx$

Solve the integral: $\int \frac{x^2}{e^x}dx$

Solve the integral: $\int 3x{e}^{x^2}dx$

Solve the integral: $\int \ln \left(3x\right)dx$