Question

How do you find the integral of `int sin^n(x)cos^m(x)` if m and n is an integer?

Answer 1

See Explanation

Explanation:

Well that would depend on what `n` and `m` are.

There are usually `4` cases to consider:

Case 1:

If `n` odd. Strip `1` sine out and convert rest to
cosines using `sin^2x = 1- cos^2x` , then use
the substitution `u = cosx` .

Case 2:

If `m` is odd, then strip `1` cosine out and convert the rest
to sines using `cos^2x=1-sin^2x` and then use
the substitution `u=sinx`

Case 3:

If both `n` and `m` are odd, we can use either method used in Case 1 & 2

Case 4:

If both `n` and `m` are even we will need to use double angle
and/or half angle formulas to reduce the
integral into something we can integrated easier.

For examples and further information check out the following resources:

Stewart Calculus: Intergrals Involving Trigonometric Functions
http://www.stewartcalculus.com/data/CALCULUS%20Concepts%20and%20Contexts/upfiles/3c3-TrigonometIntegrals_Stu.pdf

Paul's Online Math Notes:
http://tutorial.math.lamar.edu/Classes/CalcII/IntegralsWithTrig.aspx

Also for Trig Substitution:
www.math.wisc.edu/~park/Fall2011/integration/Trig%20substitution.pdf