Question

Evaluate `intcos^5x dx` using u-substitution or tabular integration?

Answer 1

`sinx - 2/3 sin^3x + 1/5 sin^5x + C`

Explanation:

We will use u-substitution and a trigonometric identity.

Recall that `cos^2 x + sin^2 x = 1`. Rearranging, it follows that `cos^2 x = 1 - sin^2 x`. Note that we can use this fact to change the integral as follows:

`int cos^5 x dx = int (cos^2x)^2 cosx dx`
`= int (1-sin^2 x)^2 cosxdx`

Make a u-substitution by letting `u = sinx`. Then `du = cosx dx`. Our integral becomes:

`int (1-u^2)^2 du = int (1 - 2u^2 + u^4) du`
`= u - 2/3 u^3 + 1/5 u^5 + C`
`= sinx - 2/3 sin^3x + 1/5sin^5x + C`

This is our final answer.