Calculus Early Transcendentals

Howard Anton, Irl Bivens, Stephen Davis

Chapter 7

PRINCIPLES OF INTEGRAL EVALUATION - all with Video Answers

Educators

Section 1

An Overview of Integration Methods

Problem 1

Use algebraic manipulation and (if necessary) $u$ -substitution
to integrate the function.
$$
\begin{array}{l}{\text { (a) } \int \frac{x+1}{x} d x=} \\ {\text { (b) } \int \frac{x+2}{x+1} d x=} \\ {\text { (c) } \int \frac{2 x+1}{x^{2}+1} d x=} \\ {\text { (d) } \int x e^{3 \ln x} d x=}\end{array}
$$

Check back soon!

Problem 1

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.
$$
\int(4-2 x)^{3} d x
$$

Geena P.

Numerade Educator

Problem 2

Use trigonometric identities and (if necessary) $u$ -substitution to integrate the function.
$$
\begin{array}{l}{\text { (a) } \int \frac{1}{\csc x} d x=} \\ {\text { (b) } \int \frac{1}{\cos ^{2} x} d x=}\end{array}
\begin{array}{l}{\text { (c) } \int\left(\cot ^{2} x+1\right) d x=} \\ {\text { (d) } \int \frac{1}{\sec x+\tan x} d x=}\end{array}
$$

Check back soon!

Problem 2

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.
$$
\int 3 \sqrt{4+2 x} d x
$$

Geena P.

Numerade Educator

Problem 3

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.
$$
\int x \sec ^{2}\left(x^{2}\right) d x
$$

Geena P.

Numerade Educator

Problem 4

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.
$$
\int 4 x \tan \left(x^{2}\right) d x
$$

Geena P.

Numerade Educator

Problem 5

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.
$$
\int \frac{\sin 3 x}{2+\cos 3 x} d x
$$

Geena P.

Numerade Educator

Problem 6

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.
$$
\int \frac{1}{9+4 x^{2}} d x
$$

Geena P.

Numerade Educator

Problem 7

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.
$$
\int e^{x} \sinh \left(e^{x}\right) d x
$$

Geena P.

Numerade Educator

Problem 8

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.
$$
\int \frac{\sec (\ln x) \tan (\ln x)}{x} d x
$$

Geena P.

Numerade Educator

Problem 9

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.
$$
\int e^{\tan x} \sec ^{2} x d x
$$

Geena P.

Numerade Educator

Problem 10

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.
$$
\int \frac{x}{\sqrt{1-x^{4}}} d x
$$

Geena P.

Numerade Educator

Problem 11

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.
$$
\int \cos ^{5} 5 x \sin 5 x d x
$$

Geena P.

Numerade Educator

Problem 12

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.
$$
\int \frac{\cos x}{\sin x \sqrt{\sin ^{2} x+1}} d x
$$

Geena P.

Numerade Educator

Problem 13

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.
$$
\int \frac{e^{x}}{\sqrt{4+e^{2 x}}} d x
$$

Geena P.

Numerade Educator

Problem 14

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.
$$
\int \frac{e^{\tan ^{-1} x}}{1+x^{2}} d x
$$

Geena P.

Numerade Educator

Problem 15

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.
$$
\int \frac{e^{\sqrt{x-1}}}{\sqrt{x-1}} d x
$$

Geena P.

Numerade Educator

Problem 16

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.
$$
\int(x+1) \cot \left(x^{2}+2 x\right) d x
$$

Geena P.

Numerade Educator

Problem 17

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.
$$
\int \frac{\cosh \sqrt{x}}{\sqrt{x}} d x
$$

Geena P.

Numerade Educator

Problem 18

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.
$$
\int \frac{d x}{x(\ln x)^{2}}
$$

Geena P.

Numerade Educator

Problem 19

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.
$$
\int \frac{d x}{\sqrt{x} 3 \sqrt{x}}
$$

Geena P.

Numerade Educator

Problem 20

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section
$$
\int \sec (\sin \theta) \tan (\sin \theta) \cos \theta d \theta
$$

Geena P.

Numerade Educator

Problem 21

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section
$$
\int \frac{\operatorname{csch}^{2}(2 / x)}{x^{2}} d x
$$

Geena P.

Numerade Educator

Problem 22

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section
$$
\int \frac{d x}{\sqrt{x^{2}-4}}
$$

Geena P.

Numerade Educator

Problem 23

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section
$$
\int \frac{e^{-x}}{4-e^{-2 x}} d x
$$

Geena P.

Numerade Educator

Problem 24

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section
$$
\int \frac{\cos (\ln x)}{x} d x
$$

Geena P.

Numerade Educator

Problem 25

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section
$$
\int \frac{e^{x}}{\sqrt{1-e^{2 x}}} d x
$$

Geena P.

Numerade Educator

Problem 26

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section
$$
\int \frac{\sinh \left(x^{-1 / 2}\right)}{x^{3 / 2}} d x
$$

Geena P.

Numerade Educator

Problem 27

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section
$$
\int \frac{x}{\csc \left(x^{2}\right)} d x
$$

Geena P.

Numerade Educator

Problem 28

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section
$$
\int \frac{e^{x}}{\sqrt{4-e^{2 x}}} d x
$$

Geena P.

Numerade Educator

Problem 29

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section
$$
\int x 4^{-x^{2}} d x
$$

Geena P.

Numerade Educator

Problem 30

Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section
$$
\int 2^{\pi x} d x
$$

Geena P.

Numerade Educator

Problem 31

(a) Evaluate the integral $\int \sin x \cos x d x$ using the substitution $u=\sin x$
(b) Evaluate the integral $\int \sin x \cos x d x$ using the identity $\sin 2 x=2 \sin x \cos x .$
(c) Explain why your answers to parts (a) and (b) are
consistent.

Geena P.

Numerade Educator

Problem 32

(a) Derive the identity
$$
\frac{\operatorname{sech}^{2} x}{1+\tanh ^{2} x}=\operatorname{sech} 2 x
$$
(b) Use the result in part (a) to evaluate $\int \operatorname{sech} x d x$.
(c) Derive the identity
$$
\operatorname{sech} x=\frac{2 e^{x}}{e^{2 x}+1}
$$
(d) Use the result in part (c) to evaluate $\int \operatorname{sech} x d x$
(e) Explain why your answers to parts (b) and (d) are
consistent.

Geena P.

Numerade Educator

Problem 33

(a) Derive the identity
$$
\frac{\sec ^{2} x}{\tan x}=\frac{1}{\sin x \cos x}
$$
(b) Use the identity $\sin 2 x=2 \sin x \cos x$ along with
the result in part (a) to evaluate $\int \csc x d x$
(c) Use the identity $\cos x=\sin [(\pi / 2)-x]$ along with
your answer to part (a) to evaluate $\int \sec x d x$

Norman A.

Numerade Educator