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Calculus

Dale Varberg, Edwin Purcell deceased, Steve Rigdon

Chapter 7

Techniques of Integration - all with Video Answers

Educators

Section 1

Basic Integration Rules

03:46

Problem 1

Perform the indicated integrations.
$$\int(x-2)^{5} d x$$

Naresh Bagrecha
Naresh Bagrecha
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07:03

Problem 2

Perform the indicated integrations.
$$\int \sqrt{3 x} d x$$

Naresh Bagrecha
Naresh Bagrecha
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06:57

Problem 3

Perform the indicated integrations.
$$\int_{0}^{2} x\left(x^{2}+1\right)^{5} d x$$

Naresh Bagrecha
Naresh Bagrecha
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07:38

Problem 4

Perform the indicated integrations.
$$\int_{0}^{1} x \sqrt{1-x^{2}} d x$$

Naresh Bagrecha
Naresh Bagrecha
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01:44

Problem 5

Perform the indicated integrations.
$$\int \frac{d x}{x^{2}+4}$$

Naresh Bagrecha
Naresh Bagrecha
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03:14

Problem 6

Perform the indicated integrations.
$$\int \frac{e^{x}}{2+e^{x}} d x$$

Naresh Bagrecha
Naresh Bagrecha
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03:45

Problem 7

Perform the indicated integrations.
$$\int \frac{x}{x^{2}+4} d x$$

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
07:47

Problem 8

Perform the indicated integrations.
$$\int \frac{2 t^{2}}{2 t^{2}+1} d t$$

Naresh Bagrecha
Naresh Bagrecha
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04:24

Problem 9

Perform the indicated integrations.
$$\int 6 z \sqrt{4+z^{2}} d z$$

Naresh Bagrecha
Naresh Bagrecha
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05:04

Problem 10

Perform the indicated integrations.
$$\int \frac{5}{\sqrt{2 t+1}} d t$$

Naresh Bagrecha
Naresh Bagrecha
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04:10

Problem 11

Perform the indicated integrations.
$$\int \frac{\tan z}{\cos ^{2} z} d z$$

Naresh Bagrecha
Naresh Bagrecha
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02:26

Problem 12

Perform the indicated integrations.
$$\int e^{\cos z} \sin z d z$$

Naresh Bagrecha
Naresh Bagrecha
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02:57

Problem 13

Perform the indicated integrations.
$$\int \frac{\sin \sqrt{t}}{\sqrt{t}} d t$$

Naresh Bagrecha
Naresh Bagrecha
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03:43

Problem 14

Perform the indicated integrations.
$$\int \frac{2 x d x}{\sqrt{1-x^{4}}}$$

Naresh Bagrecha
Naresh Bagrecha
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05:48

Problem 15

Perform the indicated integrations.
$$\int_{0}^{\pi / 4} \frac{\cos x}{1+\sin ^{2} x} d x$$

Naresh Bagrecha
Naresh Bagrecha
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07:15

Problem 16

Perform the indicated integrations.
$$\int_{0}^{3 / 4} \frac{\sin \sqrt{1-x}}{\sqrt{1-x}} d x$$

Naresh Bagrecha
Naresh Bagrecha
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07:15

Problem 17

Perform the indicated integrations.
$$\int \frac{3 x^{2}+2 x}{x+1} d x$$

Naresh Bagrecha
Naresh Bagrecha
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09:55

Problem 18

Perform the indicated integrations.
$$\int \frac{x^{3}+7 x}{x-1} d x$$

Naresh Bagrecha
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04:33

Problem 19

Perform the indicated integrations.
$$\int \frac{\sin \left(\ln 4 x^{2}\right)}{x} d x$$

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04:51

Problem 20

Perform the indicated integrations.
$$\int \frac{\sec ^{2}(\ln x)}{2 x} d x$$

Naresh Bagrecha
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04:52

Problem 21

Perform the indicated integrations.
$$\int \frac{6 e^{x}}{\sqrt{1-e^{2 x}}} d x$$

Naresh Bagrecha
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05:09

Problem 22

Perform the indicated integrations.
$$\int \frac{x}{x^{4}+4} d x$$

Naresh Bagrecha
Naresh Bagrecha
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05:47

Problem 23

Perform the indicated integrations.
$$\int \frac{3 e^{2 x}}{\sqrt{1-e^{2 x}}} d x$$

Naresh Bagrecha
Naresh Bagrecha
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03:58

Problem 24

Perform the indicated integrations.
$$\int \frac{x^{3}}{x^{4}+4} d x$$

Naresh Bagrecha
Naresh Bagrecha
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07:39

Problem 25

Perform the indicated integrations.
$$\int_{0}^{1} t 3^{t^{2}} d t$$

Naresh Bagrecha
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05:42

Problem 26

Perform the indicated integrations.
$$\int_{0}^{\pi / 6} 2^{\cos x} \sin x d x$$

Naresh Bagrecha
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05:41

Problem 27

Perform the indicated integrations.
$$\int \frac{\sin x-\cos x}{\sin x} d x$$

Naresh Bagrecha
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06:58

Problem 28

Perform the indicated integrations.
$$\int \frac{\sin (4 t-1)}{1-\sin ^{2}(4 t-1)} d t$$

Naresh Bagrecha
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03:02

Problem 29

Perform the indicated integrations.
$$\int e^{x} \sec e^{x} d x \text { Hint: See Problem } 56$$

Naresh Bagrecha
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03:13

Problem 30

Perform the indicated integrations.
$$\int e^{x} \sec ^{2}\left(e^{x}\right) d x$$

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05:22

Problem 31

Perform the indicated integrations.
$$\int \frac{\sec ^{3} x+e^{\sin x}}{\sec x} d x$$

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04:56

Problem 32

Perform the indicated integrations.
$$\int \frac{(6 t-1) \sin \sqrt{3 t^{2}-t-1}}{\sqrt{3 t^{2}-t-1}} d t$$

Naresh Bagrecha
Naresh Bagrecha
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09:12

Problem 33

Perform the indicated integrations.
$$\int \frac{t^{2} \cos \left(t^{3}-2\right)}{\sin ^{2}\left(t^{3}-2\right)} d t$$

Naresh Bagrecha
Naresh Bagrecha
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10:19

Problem 34

Perform the indicated integrations.
$$\int \frac{1+\cos 2 x}{\sin ^{2} 2 x} d x$$

Naresh Bagrecha
Naresh Bagrecha
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07:54

Problem 35

Perform the indicated integrations.
$$\int \frac{t^{2} \cos ^{2}\left(t^{3}-2\right)}{\sin ^{2}\left(t^{3}-2\right)} d t$$

Naresh Bagrecha
Naresh Bagrecha
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05:54

Problem 36

Perform the indicated integrations.
$$\int \frac{\csc ^{2} 2 t}{\sqrt{1+\cot 2 t}} d t$$

Naresh Bagrecha
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04:05

Problem 37

Perform the indicated integrations.
$$\int \frac{e^{\tan ^{-1} 2 t}}{1+4 t^{2}} d t$$

Naresh Bagrecha
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05:03

Problem 38

Perform the indicated integrations.
$$\int(t+1) e^{-t^{2}-2 t-5} d t$$

Naresh Bagrecha
Naresh Bagrecha
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07:06

Problem 39

Perform the indicated integrations.
$$\int \frac{y}{\sqrt{16-9 y^{4}}} d y$$

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03:34

Problem 40

Perform the indicated integrations.
$$\int \cosh 3 x d x$$

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Naresh Bagrecha
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03:33

Problem 41

Perform the indicated integrations.
$$\int x^{2} \sinh x^{3} d x$$

Naresh Bagrecha
Naresh Bagrecha
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07:30

Problem 42

Perform the indicated integrations.
$$\int \frac{5}{\sqrt{9-4 x^{2}}} d x$$

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
06:21

Problem 43

Perform the indicated integrations.
$$\int \frac{e^{3 t}}{\sqrt{4-e^{6 t}}} d t$$

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Naresh Bagrecha
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06:23

Problem 44

Perform the indicated integrations.
$$\int \frac{d t}{2 t \sqrt{4 t^{2}-1}}$$

Naresh Bagrecha
Naresh Bagrecha
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06:36

Problem 45

Perform the indicated integrations.
$$\int_{0}^{\pi / 2} \frac{\sin x}{16+\cos ^{2} x} d x$$

Naresh Bagrecha
Naresh Bagrecha
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07:44

Problem 46

Perform the indicated integrations.
$$\int_{0}^{1} \frac{e^{2 x}-e^{-2 x}}{e^{2 x}+e^{-2 x}} d x$$

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
06:25

Problem 47

Perform the indicated integrations.
$$\int \frac{1}{x^{2}+2 x+5} d x$$

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
05:50

Problem 48

Perform the indicated integrations.
$$\int \frac{1}{x^{2}-4 x+9} d x$$

Naresh Bagrecha
Naresh Bagrecha
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06:36

Problem 49

Perform the indicated integrations.
$$\int \frac{d x}{9 x^{2}+18 x+10}$$

Naresh Bagrecha
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06:08

Problem 50

Perform the indicated integrations.
$$\int \frac{d x}{\sqrt{16+6 x-x^{2}}}$$

Naresh Bagrecha
Naresh Bagrecha
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05:09

Problem 51

Perform the indicated integrations.
$$\int \frac{x+1}{9 x^{2}+18 x+10} d x$$

Naresh Bagrecha
Naresh Bagrecha
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05:54

Problem 52

Perform the indicated integrations.
$$\int \frac{3-x}{\sqrt{16+6 x-x^{2}}} d x$$

Naresh Bagrecha
Naresh Bagrecha
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04:55

Problem 53

Perform the indicated integrations.
$$\int \frac{d t}{t \sqrt{2 t^{2}-9}}$$

Naresh Bagrecha
Naresh Bagrecha
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07:12

Problem 54

Perform the indicated integrations.
$$\int \frac{\tan x}{\sqrt{\sec ^{2} x-4}} d x$$

Naresh Bagrecha
Naresh Bagrecha
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09:39

Problem 55

Find the length of the curve $y=\ln (\cos x)$ between $x=0$ and $x=\pi / 4$.

Geena Pullo
Geena Pullo
Numerade Educator
04:33

Problem 56

Establish the identity
$$
\sec x=\frac{\sin x}{\cos x}+\frac{\cos x}{1+\sin x}
$$
and then use it to derive the formula
$$
\int \sec x d x=\ln |\sec x+\tan x|+C
$$

Linda Hand
Linda Hand
Numerade Educator
01:28

Problem 57

Evaluate $\int_{0}^{2 \pi} \frac{x|\sin x|}{1+\cos ^{2} x} d x .$ Hint: Make the substitution $u=x-\pi$ in the definite integral and then use symmetry properties.

Rukhmani Jain
Rukhmani Jain
Numerade Educator
02:42

Problem 58

Let $R$ be the region bounded by $y=\sin x$ and $y=\cos x$ between $x=-\pi / 4$ and $x=3 \pi / 4 .$ Find the volume of the solid obtained when $R$ is revolved about $x=-\pi / 4$. Hint: Use cylindrical shells to write a single integral, make the substitution $u=x-\pi / 4,$ and apply symmetry properties.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator