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# Integrals of Trigonometric Expressions Help

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By — McGraw-Hill Professional
Updated on Aug 31, 2011

## Introduction to Integrals of Trigonometric Expressions

Trigonometric expressions arise frequently in our work, especially as a result of substitutions. In this section we develop a few examples of trigonometric integrals.

The following trigonometric identities will be particularly useful for us.

I. We have

The reason is that

cos 2x = cos2x − sin2x = [1 − sin2x ] − sin2x = 1 − 2 sin2x .

II. We have

The reason is that

cos 2x = cos2x − sin2x

sin2x = cos2x − [1 − cos2x ] = 2 cos2x − 1.

Now we can turn to some examples.

#### Example 1

Calculate the integral

∫ cos2x dx .

#### Solution 1

Of course we will use formula II . We write

#### Example 2

Calculate the integral

∫ sin3x  cos2x dx .

#### Solution 2

When sines and cosines occur together, we always focus on the odd power (when one occurs). We write

sin3x  cos2x = sinx sin2x cos2x = sin x (1 − cos2x ) cos2x

= [cos2x − cos4x ] sinx .

Then

∫ sin3x  cos2 dx = ∫ [cos2x − cos4x ] sin x dx .

A u -substitution is suggested: We let u = cos x , du = − sin x dx . Then the integral becomes

Resubstituting for the u variable, we obtain the final solution of

You Try It : Calculate the integral

∫ sin23x cos53x dx .

Calculate

#### Solution 3

Substituting

into the integrand yields

Again using formula II, we find that our integral becomes

Applying formula II one last time yields

You Try It : Calculate the integral

You Try It : Calculate the integral

Integrals involving the other trigonometric functions can also be handled with suitable trigonometric identities. We illustrate the idea with some examples that are handled with the identity

#### Example 4

Calculate

∫ tan3x sec3x dx .

#### Solution 4

Using the same philosophy about odd exponents as we did with sines and cosines, we substitute sec 2 x − 1 for tan 2 x . The result is

∫ tan x (sec2x − l) sec3 x dx .

We may regroup the terms in the integrand to obtain

∫ [sec4 x − sec2 x ] sec x tan x dx .

A u -substitution suggests itself: We let u = sec x and therefore du = sec x tan x dx . Thus our integral becomes

Resubstituting the value of u gives

Calculate

#### Solution 5

We write

Letting u = tan x and du = sec2 x dx then gives the integral

You Try It : Calculate the integral

Further techniques in the evaluation of trigonometric integrals will be explored in the exercises.

Find practice problems and solutions for these concepts at: Methods Of Integration Practice Test.

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