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

Mar 8, 2018

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 ${\sin}^{2} x = 1 - {\cos}^{2} x$ , then use
the substitution $u = \cos x$ .

Case 2:

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

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