Trigonometric Integrals

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

Videos

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

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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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