Calculus Of One Real Variable By Pheng Kim Ving
Chapter 10: Techniques Of Integration Section 10.3: Integration Of Trigonometric Fumctions

 

10.3
Integration Of Trigonometric Functions

 

 

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1. Basic Trigonometric Integrals

 

Recall from Section 10.1 Part 4 that:

 

 

For sec x:

 

 

We group the basic trigonometric integrals together here in the following box.

 

 

Example 1.1

 

Evaluate:

 

 

Solution

EOS

 

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2. Trigonometric Substitution

 

Example 2.1

 

Evaluate:

 

 

Solution

Let u = sin 3x. Then du = 3 cos 3x dx, so that cos 3x dx = (1/3) du. Thus:

 

EOS

 

The substitution u = sin 3x involves a trigonometric function, and as a consequence is called a trigonometric
substitution
.

 

Example 2.2

 

Find:

 

 

Solution

Let u = x2. Then du = 2x dx, so that x dx = (1/2) du. Thus:

 

EOS

 

The substitution u = x2 doesn't involve any trigonometric function. There's no trigonometric substitution. Integrals
involving trigonometric functions aren't always handled by using a trigonometric substitution.

 

Note that sin x2 = sin (x2), the sine of x2, not (sin x)2, denoted sin2 x, the square of sin x.

 

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Problems And Solutions

 

1. Evaluate:

 

 

Solution

 

Let u = 1 + sin x. So du = cos x dx. Thus:

 

 

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2. Calculate:

 

 

Solution

 

Let u = ln t. So du = (1/t) dt. Thus:

 

 

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3. Compute:

 

 

Solution

 

Let v = 2 + sin 3u. Then dv = 3 cos 3u du, so that cos 3u du = (1/3) dv. Thus:

 

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4. Find:

 

 

Solution

 

 

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5. a. Establish the following identities:

 

 

Solution

 

a. We have cos (x y) = cos x cos y + sin x sin y and cos (x + y) = cos x cos y sin x sin y. So
cos (x y) cos (x + y) = 2 sin x sin y, which yields:

 

 

 

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