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

1 Answer
Mar 8, 2018

Answer:

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