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 

Integration: Inverse Trigonometric Forms