 # Trigonometric Integrals

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In this lesson, we will look into some techniques of integrating powers of sine, cosine, tangent and secant. We will use trigonometric identities to integrate certain combinations of trigonometric functions.

### Odd Power of Sine or Cosine

To integrate an odd power of sine or cosine, we separate a single factor and convert the remaining even power.

If the power of cosine is odd (n = 2k + 1), save one cosine factor and use the identity sin2 x + cos2 x = 1 to express the remaining factors in terms of sine: Let u = sin x then du = cos x dx If the power of sine is odd (n = 2k + 1), save one sine factor and use the identity sin2 x + cos2 x = 1 to express the remaining factors in terms of cosine: Let u = cos x then du = – sin x dx Note: If the powers of both sine and cosine are odd, either of the above methods can be used.

Example:

Evaluate Solution:

Step 1:

Separate one cosine factor and convert the remaining cos2 x factor to an expression involving sine using the identity sin2 x + cos2 x = 1 Step 2:

Let u = sin x then du = cos x dx Example:

Evaluate Solution:

Step 1:

Separate one sine factor and convert the remaining sin 4 x factor to an expression involving cos using the identity sin2 x + cos2 x = 1 Step 2:

Let u = cos x then du = – sin x dx ### Even Powers of Sine and Cosine

If the powers of both the sine and cosine are even, use the half-angle identities Example:

Find Solution:

If we write sin2 x as 1 – cos2 x, the integral is no simpler to evaluate.

Instead, we use the half-angle formula for  Example:

Find Solution:

We write sin4 x as (sin2 x)2 and use a half-angle formula: In order to evaluate cos2 2x, we use the half angle formula  Trigonometric Integrals - Part 1 of 6
The 'cookie cutter' case of products of odds powers of sine and/or odd powers of cosine is discussed.

Trigonometric Integrals - Part 2 of 6
The 'cookie cutter' case of products of even powers of sine and even powers of cosine is discussed Trigonometric Integrals - Part 3 of 6
The 'cookie cutter' case of products of even powers of secant and powers of tangent is discussed Trigonometric Integrals - Part 4 of 6
The 'cookie cutter' case of products of odd powers of tangent and powers of secant is discussed. Trigonometric Integrals - Part 5 of 6
The 'cookie cutter' case of products of sin(mx) and cos(nx) are shown Trigonometric Integrals - Part 6 of 6
3 trigonometric integrals that do not fit any one technique are discussed.

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