10.3 Integration Of Trigonometric Functions

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

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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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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