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Chapter 7

Techniques of Integration

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Problem 1

Evaluate the integral.

$ \displaystyle \int \frac{\cos x}{1 - \sin x}\ dx $

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Problem 2

Evaluate the integral.

$ \displaystyle \int_0^1 (3x + 1)^{\sqrt{2}}\ dx $

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Problem 3

Evaluate the integral.

$ \displaystyle \int_1^4 \sqrt{y} \ln y\ dy $

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Problem 4

Evaluate the integral.

$ \displaystyle \int \frac{\sin^3 x}{\cos x}\ dx $

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Problem 5

Evaluate the integral.
$\int \frac{t}{t^{4}+2} d t$

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Problem 6

Evaluate the integral.

$ \displaystyle \int_0^1 \frac{x}{(2x + 1)^3}\ dx $

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Problem 7

Evaluate the integral.

$ \displaystyle \int_{-1}^1 \frac{e^{\arctan y}}{1 + y^2}\ dy $

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Problem 8

Evaluate the integral.

$ \displaystyle \int t \sin t \cos t\ dy $

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Problem 9

Evaluate the integral.

$ \displaystyle \int_2^4 \frac{x + 2}{x^2 + 3x - 4}\ dx $

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Problem 10

Evaluate the integral.

$ \displaystyle \int \frac{\cos (\frac{1}{x})}{x^3}\ dx $

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Problem 11

Evaluate the integral.

$ \displaystyle \int \frac{1}{x^3 \sqrt{x^2 - 1}}\ dx $

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Problem 12

Evaluate the integral.

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

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Problem 13

Evaluate the integral.

$ \displaystyle \int \sin^5 t \cos^4 t\ dt $

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Problem 14

Evaluate the integral.

$ \displaystyle \int \ln (1 + x^2)\ dx $

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Problem 15

Evaluate the integral.

$ \displaystyle \int x \sec x \tan x\ dx $

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Problem 16

Evaluate the integral.

$ \displaystyle \int_0^{\frac{\sqrt{2}}{2}} \frac{x^2}{\sqrt{1 - x^2}}\ dx $

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Problem 17

Evaluate the integral.

$ \displaystyle \int_0^\pi t \cos^2 t\ dt $

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Problem 18

Evaluate the integral.

$ \displaystyle \int_1^4 \frac{e^{\sqrt{t}}}{\sqrt{t}}\ dt $

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Problem 19

Evaluate the integral.

$ \displaystyle \int e^{x + e^x}\ dx $

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Problem 20

Evaluate the integral.

$ \displaystyle \int e^2\ dx $

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Problem 21

Evaluate the integral.

$ \displaystyle \int \arctan \sqrt{x}\ dx $

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Problem 22

Evaluate the integral.

$ \displaystyle \int \frac{\ln x}{x \sqrt{1 + (\ln x)^2}}\ dx $

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Problem 23

Evaluate the integral.

$ \displaystyle \int_0^1 (1 + \sqrt{x})^8\ dx $

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Problem 24

Evaluate the integral.

$ \displaystyle \int (1 + \tan x)^2 \sec x\ dx $

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Problem 25

Evaluate the integral.

$ \displaystyle \int_0^1 \frac{1 + 12t}{1 + 3t}\ dt $

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Problem 26

Evaluate the integral.

$ \displaystyle \int_0^1 \frac{3x^2 + 1}{x^3 + x^2 + x + 1}\ dx $

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Problem 27

Evaluate the integral.

$ \displaystyle \int \frac{dx}{1+ e^x} $

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Problem 28

Evaluate the integral.

$ \displaystyle \int \sin \sqrt{at}\ dt $

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Problem 29

Evaluate the integral.

$ \displaystyle \int \ln (x + \sqrt{x^2 - 1})\ dx $

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Problem 30

Evaluate the integral.

$ \displaystyle \int_{-1}^2 \mid e^x - 1 \mid dx $

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Problem 31

Evaluate the integral.

$ \displaystyle \int \sqrt{\frac{1 + x}{1 - x}}\ dx $

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Problem 32

Evaluate the integral.

$ \displaystyle \int_1^3 \frac{e^{\frac{3}{x}}}{x^2}\ dx $

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Problem 33

Evaluate the integral.

$ \displaystyle \int \sqrt{3 - 2x - x^2}\ dx $

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Problem 34

Evaluate the integral.

$ \displaystyle \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{1 + 4 \cot x}{4 - \cot x}\ dx $

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Problem 35

Evaluate the integral.

$ \displaystyle \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{x}{1 + \cos^2 x}\ dx $

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Problem 36

Evaluate the integral.

$ \displaystyle \int \frac{1 + \sin x}{1 + \cos x}\ dx $

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Problem 37

Evaluate the integral.

$ \displaystyle \int_0^{\frac{\pi}{4}} \tan^3 \theta \sec^2 \theta\ d \theta $

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Problem 38

Evaluate the integral.

$ \displaystyle \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin \theta \cot \theta}{\sec \theta}\ d \theta $

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Problem 39

Evaluate the integral.

$ \displaystyle \int \frac{\sec \theta \tan \theta}{\sec^2 \theta - \sec \theta}\ d \theta $

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Problem 40

Evaluate the integral.

$ \displaystyle \int_0^\pi \sin 6x \cos 3x\ dx $

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Problem 41

Evaluate the integral.

$ \displaystyle \int \theta \tan^2 \theta\ d \theta $

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Problem 42

Evaluate the integral.

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

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Problem 43

Evaluate the integral.

$ \displaystyle \int \frac{\sqrt{x}}{1 + x^3}\ dx $

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Problem 44

Evaluate the integral.

$ \displaystyle \int \sqrt{1 + e^x}\ dx $

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Problem 45

Evaluate the integral.

$ \displaystyle \int x^5 e^{-x^3}\ dx $

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Problem 46

Evaluate the integral.

$ \displaystyle \int \frac{(x - 1) e^x}{x^2}\ dx $

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Problem 47

Evaluate the integral.

$ \displaystyle \int x^3 (x - 1)^{-4}\ dx $

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Problem 48

Evaluate the integral.

$ \displaystyle \int_0^1 x \sqrt{2 - \sqrt{1 - x^2}}\ dx $

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Problem 49

Evaluate the integral.

$ \displaystyle \int \frac{1}{x \sqrt{4x + 1}}\ dx $

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Problem 50

Evaluate the integral.

$ \displaystyle \int \frac{1}{x^2 \sqrt{4x + 1}}\ dx $

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Problem 51

Evaluate the integral.

$ \displaystyle \int \frac{1}{x \sqrt{4x^2 + 1}}\ dx $

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Problem 52

Evaluate the integral.

$ \displaystyle \int \frac{dx}{x (x^4 + 1)} $

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Problem 53

Evaluate the integral.

$ \displaystyle \int x^2 \sinh mx\ dx $

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Problem 54

Evaluate the integral.

$ \displaystyle \int (x + \sin x)^2\ dx $

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Problem 55

Evaluate the integral.

$ \displaystyle \int \frac{dx}{x + x \sqrt{x}}\ $

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Problem 56

Evaluate the integral.

$ \displaystyle \int \frac{dx}{\sqrt{x} + x \sqrt{x}}\ $

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Problem 57

Evaluate the integral.

$ \displaystyle \int x^3 \sqrt{x + c}\ dx $

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Problem 58

Evaluate the integral.

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

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Problem 59

Evaluate the integral.

$ \displaystyle \int \frac{dx}{x^4 - 16} $

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Problem 60

Evaluate the integral.

$ \displaystyle \int \frac{dx}{x^2 \sqrt{4x^2 - 1}} $

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Problem 61

Evaluate the integral.

$ \displaystyle \int \frac{d \theta}{1 + \cos \theta} $

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Problem 62

Evaluate the integral.

$ \displaystyle \int \frac{d \theta}{1 + \cos^2 \theta} $

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Problem 63

Evaluate the integral.

$ \displaystyle \int \sqrt{x} e^{\sqrt{x}}\ dx $

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Problem 64

Evaluate the integral.

$ \displaystyle \int \frac{1}{\sqrt{\sqrt{x} + 1}}\ dx $

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Problem 65

Evaluate the integral.

$ \displaystyle \int \frac{\sin 2x}{1 + \cos^4 x}\ dx $

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Problem 66

Evaluate the integral.

$ \displaystyle \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{\ln (\tan x)}{\sin x \cos x}\ dx $

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Problem 67

Evaluate the integral.

$ \displaystyle \int \frac{1}{\sqrt{x + 1} + \sqrt{x}}\ dx $

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Problem 68

Evaluate the integral.

$ \displaystyle \int \frac{x^2}{x^6 + 3x^3 + 2}\ dx $

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Problem 69

Evaluate the integral.

$ \displaystyle \int_1^{\sqrt{3}} \frac{\sqrt{1 + x^2}}{x^2}\ dx $

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Problem 70

Evaluate the integral.

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

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Problem 71

Evaluate the integral.

$ \displaystyle \int \frac{e^{2x}}{1 + e^x}\ dx $

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Problem 72

Evaluate the integral.

$ \displaystyle \int \frac{\ln (x + 1)}{x^2}\ dx $

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Problem 73

Evaluate the integral.

$ \displaystyle \int \frac{x + \arcsin x}{\sqrt{1 - x^2}}\ dx $

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Problem 74

Evaluate the integral.

$ \displaystyle \int \frac{4^x + 10^x}{2^x}\ dx $

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Problem 75

Evaluate the integral.

$ \displaystyle \int \frac{dx}{x \ln x - x} $

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Problem 76

Evaluate the integral.

$ \displaystyle \int \frac{x^2}{\sqrt{x^2 + 1}}\ dx $

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Problem 77

Evaluate the integral.

$ \displaystyle \int \frac{xe^x}{\sqrt{1 + e^x}}\ dx $

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Problem 78

Evaluate the integral.

$ \displaystyle \int \frac{1 + \sin x}{1 - \sin x}\ dx $

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Problem 79

Evaluate the integral.

$ \displaystyle \int x \sin^2 x \cos x\ dx $

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Problem 80

Evaluate the integral.

$ \displaystyle \int \frac{\sec x \cos 2x}{\sin x + \sec x}\ dx $

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Problem 81

Evaluate the integral.

$ \displaystyle \int \sqrt{1 - \sin x}\ dx $

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Problem 82

Evaluate the integral.

$ \displaystyle \int \frac{\sin x \cos x}{\sin^4 x + \cos^4 x}\ dx $

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Problem 83

The functions $ y = e^{x^2} $ and $ y = x^2 e^{x^2} $ don't have elementary antiderivatives, but $ y = (2x^2 + 1) e^{x^2} $ does. Evaluate $ \int (2x^2 + 1) e^{x^2}\ dx $.

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Problem 84

We know that $ F(x) = \displaystyle \int_0^x e^{e^t}\ dt $ is a continuous function by FTC1, though it is not an elementary function. The functions
$ \displaystyle \int \frac{e^x}{x}\ dx $ and $ \displaystyle \int \frac{1}{\ln x}\ dx $
are not elementary either, but they can be expressed in terms of $ F $. Evaluate the following integrals in terms of $ F $.
$ \displaystyle \int_1^2 \frac{e^x}{x}\ dx $ and $ \displaystyle \int_2^3 \frac{1}{\ln x}\ dx $

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