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Calculus

Earl W. Swokowski

Chapter 9

Techniques of Integration - all with Video Answers

Educators

Section 1

Integration by Parts

00:35

Problem 1

Evaluate the integral.
$\int x e^{-x} d x$

Fuzail Shakir
Fuzail Shakir
Numerade Educator
01:25

Problem 2

Evaluate the integral.
$\int x \sin x d x$

Steven Clarke
Steven Clarke
Numerade Educator
04:54

Problem 3

Evaluate the integral.
$\int x^{2} e^{3 x} d x$

Steven Clarke
Steven Clarke
Numerade Educator
02:34

Problem 4

Evaluate the integral.
$\int x^{2} \sin 4 x d x$

Yiyang Wang
Yiyang Wang
Numerade Educator
01:08

Problem 5

Evaluate the integral.
$\int x^{2} \sin 4 x d x$

Yiyang Wang
Yiyang Wang
Numerade Educator
01:35

Problem 6

Evaluate the integral.
$\int x e^{-2 x} d x$

Amy Jiang
Amy Jiang
Numerade Educator
02:33

Problem 7

Evaluate the integral.
$\int x \sec x \tan x d x$

Nafis Fuad
Nafis Fuad
Numerade Educator
02:53

Problem 8

Evaluate the integral.
$\int x \csc ^{2} 3 x d x$

Steven Clarke
Steven Clarke
Numerade Educator
01:01

Problem 9

Evaluate the integral.
$\int x^{2} \cos x d x$

Amy Jiang
Amy Jiang
Numerade Educator
01:54

Problem 10

Evaluate the integral.
$\int x^{3} e^{-x} d x$

Yiyang Wang
Yiyang Wang
Numerade Educator
07:03

Problem 11

Evaluate the integral.
$\int \tan ^{-1} x d x$

Perry Roeder
Perry Roeder
Numerade Educator
02:36

Problem 12

Evaluate the integral.
$\int \sin ^{-1} x d x$

Robert Daugherty
Robert Daugherty
Numerade Educator
03:15

Problem 13

Evaluate the integral.
$\int \sqrt{x} \ln x d x$

Lucas Gagne
Lucas Gagne
Numerade Educator
00:51

Problem 14

Evaluate the integral.
$\int x^{2} \ln x d x$

Linda Hand
Linda Hand
Numerade Educator
01:44

Problem 15

Evaluate the integral.
$\int x \csc ^{2} x d x$

Steven Clarke
Steven Clarke
Numerade Educator
07:03

Problem 16

Evaluate the integral.
$\int x \tan ^{-1} x d x$

Perry Roeder
Perry Roeder
Numerade Educator
06:45

Problem 17

Evaluate the integral.
$\int e^{-x} \sin x d x$

Perry Roeder
Perry Roeder
Numerade Educator
10:54

Problem 18

Evaluate the integral.
$\int e^{3 x} \cos 2 x d x$

Perry Roeder
Perry Roeder
Numerade Educator
02:33

Problem 19

Evaluate the integral.
$\int \sin x \ln \cos x d x$

Steven Clarke
Steven Clarke
Numerade Educator
02:50

Problem 20

Evaluate the integral.
$\int_{0}^{1} x^{3} e^{-x^{2}} d x$

Yiyang Wang
Yiyang Wang
Numerade Educator
02:00

Problem 21

Evaluate the integral.
$\int \csc ^{3} x d x$

Yiyang Wang
Yiyang Wang
Numerade Educator
04:09

Problem 22

Evaluate the integral.
$\int \sec ^{5} x d x$

Geena Pullo
Geena Pullo
Numerade Educator
02:16

Problem 23

Evaluate the integral.
$\int_{0}^{1} \frac{x^{3}}{\sqrt{x^{2}+1}} d x$

Yiyang Wang
Yiyang Wang
Numerade Educator
02:22

Problem 24

Evaluate the integral.
$\int \sin \ln x d x$

Yiyang Wang
Yiyang Wang
Numerade Educator
03:00

Problem 25

Evaluate the integral.
$\int_{0}^{\pi / 2} x \sin 2 x d x$

Steven Clarke
Steven Clarke
Numerade Educator
02:57

Problem 26

Evaluate the integral.
$\int x \sec ^{2} 5 x d x$

Steven Clarke
Steven Clarke
Numerade Educator
03:44

Problem 27

Evaluate the integral.
$\int x(2 x+3)^{99} d x$

Steven Clarke
Steven Clarke
Numerade Educator
02:01

Problem 28

Evaluate the integral.
$\int \frac{x^{5}}{\sqrt{1-x^{3}}} d x$

Yiyang Wang
Yiyang Wang
Numerade Educator
02:43

Problem 29

Evaluate the integral.
$\int e^{4 x} \sin 5 x d x$

Yiyang Wang
Yiyang Wang
Numerade Educator
01:15

Problem 30

Evaluate the integral.
$\int x^{3} \cos \left(x^{2}\right) d x$

Yiyang Wang
Yiyang Wang
Numerade Educator
02:05

Problem 31

Evaluate the integral.
$\int(\ln x)^{2} d x$

Amy Jiang
Amy Jiang
Numerade Educator
01:21

Problem 32

Evaluate the integral.
$\int x 2^{x} d x$

Yiyang Wang
Yiyang Wang
Numerade Educator
02:00

Problem 33

Evaluate the integral.
$\int x^{3} \sinh x d x$

Yiyang Wang
Yiyang Wang
Numerade Educator
01:08

Problem 34

Evaluate the integral.
$\int(x+4) \cosh 4 x d x$

Yiyang Wang
Yiyang Wang
Numerade Educator
01:45

Problem 35

Evaluate the integral.
$\int \cos \sqrt{x} d x$

Harmender Singh Yadav
Harmender Singh Yadav
Numerade Educator
06:01

Problem 36

Evaluate the integral.
$\int \tan ^{-1} 3 x d x$

Perry Roeder
Perry Roeder
Numerade Educator
02:49

Problem 37

Evaluate the integral.
$\int \cos ^{-1} x d x$

Nafis Fuad
Nafis Fuad
Numerade Educator
01:35

Problem 38

Evaluate the integral.
$\int(x+1)^{10}(x+2) d x$

Yiyang Wang
Yiyang Wang
Numerade Educator
01:24

Problem 39

Use integration by parts to derive the reduction formula.
$$\int x^{m} e^{x} d x=x^{m} e^{x}-m \int x^{m-1} e^{x} d x$$

Yiyang Wang
Yiyang Wang
Numerade Educator
02:06

Problem 40

Use integration by parts to derive the reduction formula.
$$\int x^{m} \sin x d x=-x^{m} \cos x+m \int x^{m-1} \cos x d x$$

Steven Clarke
Steven Clarke
Numerade Educator
02:21

Problem 41

Use integration by parts to derive the reduction formula.
$$\int(\ln x)^{m} d x=x(\ln x)^{m}-m \int(\ln x)^{m-1} d x$$

Steven Clarke
Steven Clarke
Numerade Educator
03:33

Problem 42

Use integration by parts to derive the reduction formula.
$$\begin{array}{l}\int \sec ^{m} x d x=\frac{\sec ^{m-2} x \tan x}{m-1}+\frac{m-2}{m-1} \int \sec ^{m-2} x d x \\\text { for } m \neq 1 .\end{array}$$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
02:23

Problem 43

Use Exercise 39 to evaluate $\int x^{5} e^{x} d x$.

Yiyang Wang
Yiyang Wang
Numerade Educator
02:19

Problem 44

Use Exercise 41 to evaluate $\int(\ln x)^{4} d x$.

Ashley Boni
Ashley Boni
Numerade Educator
03:01

Problem 45

If $f(x)=\sin \sqrt{x},$ find the area of the region under the graph of $f$ from $x=0$ to $x=\pi^{2}$.

Yiyang Wang
Yiyang Wang
Numerade Educator
02:33

Problem 46

The region between the graph of $y=x \sqrt{\sin x}$ and the $x$ -axis from $x=0$ to $x=\pi / 2$ is revolved about the $x$ -axis. Find the volume of the resulting solid.

Yiyang Wang
Yiyang Wang
Numerade Educator
02:34

Problem 47

The region bounded by the graphs of $y=\ln x, y=0$. and $x=e$ is revolved about the $y$ -axis. Find the volume of the resulting solid.

Yiyang Wang
Yiyang Wang
Numerade Educator
02:37

Problem 48

Suppose the force $f(x)$ acting at the point with coordinate $x$ on a coordinate line $l$ is given by $f(x)=$ $x^{5} \sqrt{x^{3}+1}$. Find the work done in moving an object from $x=0$ to $x=1$.

Donald Albin
Donald Albin
Numerade Educator
02:39

Problem 49

Find the centroid of the region bounded by the graphs of the equations $y=e^{x}, y=0, x=0,$ and $x=\ln 3$.

Lucas Finney
Lucas Finney
Numerade Educator
01:56

Problem 50

The velocity (at time $t$ ) of a point moving along a coordinate line is $t / e^{2 t} \mathrm{ft} / \mathrm{sec} .$ If the point is at the origin at $t=0,$ find its position at time $t$.

Lucas Finney
Lucas Finney
Numerade Educator
02:11

Problem 51

When applying the integration by parts formula (9.1), show that if, after choosing $d v$, we use $v+C$ in place of $v$, the same result is obtained.

Uma Kumari
Uma Kumari
Numerade Educator
01:37

Problem 52

In Section 6.3 the discussion of finding volumes by means of cylindrical shells was incomplete because we did not show that the same result is obtained if the disk method is also applicable. Use integration by parts to prove that if $f$ is differentiable and either $f^{\prime}(x)>0$ on $[a, b]$ or $f^{\prime}(x)<0$ on $[a, b],$ and if $V$ is the volume of the solid.

Yiming Zhang
Yiming Zhang
Numerade Educator
01:51

Problem 53

Discuss the following use of Formula (9.1): Given $\int(1 / x) d x,$ let $d v=d x$ and $u=1 / x$ so that $v=x$ and $d u=\left(-1 / x^{2}\right) d x .$ Hence or
$$
\begin{array}{c}
\int \frac{1}{x} d x=\left(\frac{1}{x}\right) x-\int x\left(-\frac{1}{x^{2}}\right) d x \\
\int \frac{1}{x} d x=1+\int \frac{1}{x} d x
\end{array}
$$
Consequently, $0=1$

Sirat Shah
Sirat Shah
Numerade Educator
02:04

Problem 54

If $u=f(x)$ and $v=g(x),$ prove that the analogue of Formula (9.1) for definite integrals is
$$
\int_{a}^{b} u d v=[u v]_{a}^{b}-\int_{a}^{b} v d u
$$
for values $a$ and $b$ of $x$.

Rukhmani Jain
Rukhmani Jain
Numerade Educator