Integrating the product of the power of sine by the power of cosine

Integrating a function of the product of sine by cosine: f(x) = sin^mxcos^nx ( read sine  exponent m x by cos exponent n x)

Method

1. If m is odd let u = cosx
2. If n is odd let u = sinx
3. If m and n are even use identities to reduce the power of sine and cosine

Example I

Solve ∫sin³xcos⁴xdx

 Since m is odd let u = cox then du =- sinxdx  dx = -du/sinx

Let's substitute cox and dx in the integral

∫sin³xcos⁴xdx = ∫sin³xu⁴.-du/sinx

Let's simplify by sinx:

∫sin³xcos⁴xdx = - ∫sin²xu⁴du

In order to have the integral as a function of u let's express sin²x as an expression of cosx

sin²x = 1-cos²x = 1-u²

Let's substitute sin²x

∫sin³xcos⁴xdx = -∫(1-u²)u⁴du

                     = -∫(u⁴-u⁶)du

                     = -(u⁵/5-u⁷/7 +C

                     = -u⁵/5-u⁷/7 +C

                     =  -cos⁵x/5-cos⁷x/7 +C ( by substituting u by cosx)

Example II

Solve ∫sin²xcos²xdx.

We have m and n even. We use identities to reduce power.

 sin²xcos²xdx. = (1 - cos2x)/2.(1 + cos2x)/2 = 1 - cos²2x/4 = sin²2x/4

Let's reduce the power of sin²2x

 sin²2x = (1 - cos4x)/2

 sin²2x/4 = (1 - cos4x)/8 = 1/8 - 1/8cos4x

 ∫sin²xcos²xdx = ∫(1/8 - 1/8cos4x)dx

                      =1/8 ∫dx - 1/8 ∫cos4xdx

                      = 1/8x - 1/8sin4x + C

Practice

Solve

1) ∫sin⁴xcos³x
2) ∫sin⁴xcos⁴x
.