Calculus Early Transcendentals

William Briggs, Lyle Cochran, Bernard Gillett

Chapter 7 Integration Techniques - all with Video Answers

Educators

Section 1

Basic Approaches

00:52

Problem 1

What change of variables would you use for the integral $\int(4-7 x)^{-6} d x ?$

Robert Daugherty

Numerade Educator

00:37

Problem 2

Before integrating, how would you rewrite the integrand of $\int\left(x^{4}+2\right)^{2} d x ?$

Robert Daugherty

Numerade Educator

00:45

Problem 3

What trigonometric identity is useful in evaluating $\int \sin ^{2} x d x ?$

Ramzi Deek

Numerade Educator

02:04

Problem 4

Describe a first step in integrating $\int \frac{x^{3}-2 x+4}{x-1} d x.$

Robert Daugherty

Numerade Educator

00:53

Problem 5

Describe a first step in integrating $\int \frac{10}{x^{2}-4 x+5} d x.$

Steven Clarke

Numerade Educator

01:00

Problem 6

Describe a first step in integrating $\int \frac{x^{10}-2 x^{4}+10 x^{2}+1}{3 x^{3}} d x.$

Robert Daugherty

Numerade Educator

02:00

Problem 7

Evaluate the following integrals.
$$\int \frac{d x}{(3-5 x)^{4}}$$

Ramzi Deek

Numerade Educator

01:37

Problem 8

Evaluate the following integrals.
$$\int(9 x-2)^{-3} d x$$

Robert Daugherty

Numerade Educator

03:09

Problem 9

Evaluate the following integrals.
$$\int_{0}^{3 \pi / 8} \sin \left(2 x-\frac{\pi}{4}\right) d x$$

Ramzi Deek

Numerade Educator

01:19

Problem 10

Evaluate the following integrals.
$$\int e^{3-4 x} d x$$

Ramzi Deek

Numerade Educator

01:16

Problem 11

Evaluate the following integrals.
$$\int \frac{\ln 2 x}{x} d x$$

Ramzi Deek

Numerade Educator

02:49

Problem 12

Evaluate the following integrals.
$$\int_{-5}^{0} \frac{d x}{\sqrt{4-x}}$$

Robert Daugherty

Numerade Educator

01:10

Problem 13

Evaluate the following integrals.
$$\int \frac{e^{x}}{e^{x}+1} d x$$

Ramzi Deek

Numerade Educator

01:00

Problem 14

Evaluate the following integrals.
$$\int \frac{e^{2 \sqrt{y}+1}}{\sqrt{y}} d y$$

Linh Vu

Numerade Educator

02:55

Problem 15

Evaluate the following integrals.
$$\int \frac{e^{x}}{e^{x}-2 e^{-x}} d x$$

Robert Daugherty

Numerade Educator

01:24

Problem 16

Evaluate the following integrals.
$$\int \frac{e^{2 z}}{e^{2 z}-4 e^{-z}} d z$$

Steven Clarke

Numerade Educator

02:24

Problem 17

Evaluate the following integrals.
$$\int_{1}^{e^{2}} \frac{\ln ^{2}\left(x^{2}\right)}{x} d x$$

Ramzi Deek

Numerade Educator

01:30

Problem 18

Evaluate the following integrals.
$$\int \frac{\sin ^{3} x}{\cos ^{5} x} d x$$

Amit Srivastava

Numerade Educator

02:02

Problem 19

Evaluate the following integrals.
$$\int \frac{\cos ^{4} x}{\sin ^{6} x} d x$$

Robert Daugherty

Numerade Educator

03:37

Problem 20

Evaluate the following integrals.
$$\int_{0}^{2} \frac{x(3 x+2)}{\sqrt{x^{3}+x^{2}+4}} d x$$

Robert Daugherty

Numerade Educator

02:18

Problem 21

Evaluate the following integrals.
$$\int \frac{d x}{x^{-1}+1}$$

Ramzi Deek

Numerade Educator

02:52

Problem 22

Evaluate the following integrals.
$$\int \frac{d y}{y^{-1}+y^{-3}}$$

Ramzi Deek

Numerade Educator

04:08

Problem 23

Evaluate the following integrals.
$$\int \frac{x+2}{x^{2}+4} d x$$

Ramzi Deek

Numerade Educator

02:21

Problem 24

Evaluate the following integrals.
$$\int_{4}^{9} \frac{x^{5 / 2}-x^{1 / 2}}{x^{3 / 2}} d x$$

Robert Daugherty

Numerade Educator

03:40

Problem 25

Evaluate the following integrals.
$$\int \frac{\sin t+\tan t}{\cos ^{2} t} d t$$

Robert Daugherty

Numerade Educator

01:07

Problem 26

Evaluate the following integrals.
$$\int \frac{4+e^{-2 x}}{e^{3 x}} d x$$

Ramzi Deek

Numerade Educator

02:59

Problem 27

Evaluate the following integrals.
$$\int \frac{2-3 x}{\sqrt{1-x^{2}}} d x$$

Robert Daugherty

Numerade Educator

02:33

Problem 28

Evaluate the following integrals.
$$\int \frac{3 x+1}{\sqrt{4-x^{2}}} d x$$

Robert Daugherty

Numerade Educator

01:03

Problem 29

Evaluate the following integrals.
$$\int \frac{x+2}{x+4} d x$$

Robert Daugherty

Numerade Educator

02:49

Problem 30

Evaluate the following integrals.
$$\int_{2}^{4} \frac{x^{2}+2}{x-1} d x$$

Ramzi Deek

Numerade Educator

01:58

Problem 31

Evaluate the following integrals.
$$\int \frac{t^{3}-2}{t+1} d t$$

Robert Daugherty

Numerade Educator

02:14

Problem 32

Evaluate the following integrals.
$$\int \frac{6-x^{4}}{x^{2}+4} d x$$

Robert Daugherty

Numerade Educator

02:00

Problem 33

Evaluate the following integrals.
$$\int \frac{d \theta}{1+\sin \theta}$$

Ramzi Deek

Numerade Educator

05:14

Problem 34

Evaluate the following integrals.
$$\int_{0}^{2} \frac{x}{x^{2}+4 x+8} d x$$

Patrick Vaughn

Numerade Educator

02:10

Problem 35

Evaluate the following integrals.
$$\int \frac{d \theta}{\sqrt{27-6 \theta-\theta^{2}}}$$

Robert Daugherty

Numerade Educator

02:09

Problem 36

Evaluate the following integrals.
$$\int \frac{x}{x^{4}+2 x^{2}+1} d x$$

Robert Daugherty

Numerade Educator

02:00

Problem 37

Evaluate the following integrals.
$$\int \frac{d \theta}{1+\sin \theta}$$

Ramzi Deek

Numerade Educator

01:09

Problem 38

Evaluate the following integrals.
$$\int \frac{1-x}{1-\sqrt{x}} d x$$

Robert Daugherty

Numerade Educator

03:37

Problem 39

Evaluate the following integrals.
$$\int \frac{d x}{\sec x-1}$$

Robert Daugherty

Numerade Educator

05:10

Problem 40

Evaluate the following integrals.
$$\int \frac{d \theta}{1-\csc \theta}$$

Robert Daugherty

Numerade Educator

06:19

Determine whether the following statements are true and give an explanation or counterexample.
a. $\int \frac{3}{x^{2}+4} d x=\int \frac{3}{x^{2}} d x+\int \frac{3}{4} d x$
b. Long division simplifies the evaluation of the integral $\int \frac{x^{3}+2}{3 x^{4}+x} d x$
c. $\int \frac{d x}{\sin x+1}=\ln |\sin x+1|+C$
d. $\int \frac{d x}{e^{x}}=\ln e^{x}+C$

Robert Daugherty

Numerade Educator

04:25

Problem 42

Use the approaches discussed in this section to evaluate the following integrals.
$$\int_{4}^{9} \frac{d x}{1-\sqrt{x}}$$

Robert Daugherty

Numerade Educator

05:53

Problem 43

Use the approaches discussed in this section to evaluate the following integrals.
$$\int_{-1}^{0} \frac{x}{x^{2}+2 x+2} d x$$

Robert Daugherty

Numerade Educator

07:03

Problem 44

Use the approaches discussed in this section to evaluate the following integrals.
$$\int_{0}^{1} \sqrt{1+\sqrt{x}} d x$$

Robert Daugherty

Numerade Educator

01:26

Problem 45

Use the approaches discussed in this section to evaluate the following integrals.
$$\int \sin x \sin 2 x d x$$

Robert Daugherty

Numerade Educator

02:37

Problem 46

Use the approaches discussed in this section to evaluate the following integrals.
$$\int_{0}^{\pi / 2} \sqrt{1+\cos 2 x} d x$$

Robert Daugherty

Numerade Educator

02:28

Problem 47

Use the approaches discussed in this section to evaluate the following integrals.
$$\int \frac{d x}{x^{1 / 2}+x^{3 / 2}}$$

Robert Daugherty

Numerade Educator

04:25

Problem 48

Use the approaches discussed in this section to evaluate the following integrals.
$$\int_{0}^{1} \frac{d p}{4-\sqrt{p}}$$

Robert Daugherty

Numerade Educator

05:01

Problem 49

Use the approaches discussed in this section to evaluate the following integrals.
$$\int \frac{x-2}{x^{2}+6 x+13} d x$$

Robert Daugherty

Numerade Educator

04:01

Problem 50

Use the approaches discussed in this section to evaluate the following integrals.
$$\int_{0}^{\pi / 4} 3 \sqrt{1+\sin 2 x} d x$$

Robert Daugherty

Numerade Educator

01:37

Problem 51

Use the approaches discussed in this section to evaluate the following integrals.
$$\int \frac{e^{x}}{e^{2 x}+2 e^{x}+1} d x$$

Robert Daugherty

Numerade Educator

03:52

Problem 52

Use the approaches discussed in this section to evaluate the following integrals.
$$\int_{0}^{\pi / 8} \sqrt{1-\cos 4 x} d x$$

Robert Daugherty

Numerade Educator

01:52

Problem 53

Use the approaches discussed in this section to evaluate the following integrals.
$$\int_{1}^{3} \frac{2}{x^{2}+2 x+1} d x$$

Robert Daugherty

Numerade Educator

02:28

Problem 54

Use the approaches discussed in this section to evaluate the following integrals.
$$\int_{0}^{2} \frac{2}{s^{3}+3 s^{2}+3 s+1} d s$$

Robert Daugherty

Numerade Educator

03:39

Problem 55

Different substitutions
a. Evaluate $\int \tan x \sec ^{2} x d x$ using the substitution $u=\tan x$
b. Evaluate $\int \tan x \sec ^{2} x d x$ using the substitution $u=\sec x$
c. Reconcile the results in parts (a) and (b).

Ramzi Deek

Numerade Educator

03:51

Problem 56

Different methods
a. Evaluate $\int \cot x \csc ^{2} x d x$ using the substitution $u=\cot x$
b. Evaluate $\int \cot x \csc ^{2} x d x$ using the substitution $u=\csc x$
c. Reconcile the results in parts (a) and (b).

Robert Daugherty

Numerade Educator

05:40

Problem 57

Different methods
a. Evaluate $\int \frac{x^{2}}{x+1} d x$ using the substitution $u=x+1$
b. Evaluate $\int \frac{x^{2}}{x+1} d x$ after first performing long division
on the integrand.
c. Reconcile the results in parts (a) and (b).

Robert Daugherty

Numerade Educator

05:32

Problem 58

Different substitutions
a. Show that $\int \frac{d x}{\sqrt{x-x^{2}}}=\sin ^{-1}(2 x-1)+C$ using either
$u=2 x-1$ or $u=x-\frac{1}{2}$
b. Show that $\int \frac{d x}{\sqrt{x-x^{2}}}=2 \sin ^{-1} \sqrt{x}+C$ using $u=\sqrt{x}$
c. Prove the identity $2 \sin ^{-1} \sqrt{x}-\sin ^{-1}(2 x-1)=\frac{\pi}{2}$
(Source: The College Mathematics Journal 32, 5, Nov 2001)

Patrick Vaughn

Numerade Educator

09:54

Problem 59

Area of a region between curves Find the area of the region bounded by the curves $y=\frac{x^{2}}{x^{3}-3 x}$ and $y=\frac{1}{x^{3}-3 x}$ on the interval $[2,4] .$

Robert Daugherty

Numerade Educator

06:46

Problem 60

Area of a region between curves Find the area of the entire region bounded by the curves $y=\frac{x^{3}}{x^{2}+1}$ and $y=\frac{8 x}{x^{2}+1}.$

Patrick Vaughn

Numerade Educator

02:35

Problem 61

Volumes of solids Consider the region $R$ bounded by the graph of $f(x)=\sqrt{x^{2}+1}$ and the $x$ -axis on the interval [0,2].
a. Find the volume of the solid formed when $R$ is revolved about the $x$ -axis.
b. Find the volume of the solid formed when $R$ is revolved about the $y$ -axis.

Patrick Vaughn

Numerade Educator

06:09

Problem 62

Volumes of solids Consider the region $R$ bounded by the graph of $f(x)=\frac{1}{x+2}$ and the $x$ -axis on the interval [0,3].
a. Find the volume of the solid formed when $R$ is revolved about the $x$ -axis.
b. Find the volume of the solid formed when $R$ is revolved about the $y$ -axis.

Patrick Vaughn

Numerade Educator

02:00

Problem 63

Arc length Find the length of the curve $y=x^{5 / 4}$ on the interval [0,1] . (Hint: Write the arc length integral and let $u^{2}=1+\left(\frac{5}{4}\right)^{2} \sqrt{x}.$

Harshita Goel

Numerade Educator

01:59

Problem 64

Surface area Find the area of the surface generated when the region bounded by the graph of $y=e^{x}+\frac{1}{4} e^{-x}$ on the interval $[0, \ln 2]$ is revolved about the $x$ -axis.

Ethan Feldman

Numerade Educator

03:56

Problem 65

Surface area Let $f(x)=\sqrt{x+1} .$ Find the area of the surface generated when the region bounded by the graph of $f$ on the interval [0,1] is revolved about the $x$ -axis.

Patrick Vaughn

Numerade Educator

09:18

Problem 66

Skydiving A skydiver in free fall subject to gravitational acceleration and air resistance has a velocity given by $v(t)=v_{T}\left(\frac{e^{a t}-1}{e^{a t}+1}\right),$ where $v_{T}$ is the terminal velocity and $a>0$ is a physical constant. Find the distance that the skydiver falls after $t$ seconds, which is $d(t)=\int_{0}^{t} v(y) d y.$

Patrick Vaughn

Numerade Educator

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