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Problem with trig integral

  1. Apr 13, 2014 #1
    1. The problem statement, all variables and given/known data

    Evaluate the integral.

    2. Relevant equations

    [tex] \int sin^2(\pi x) cos^5 (\pi x) dx [/tex]

    3. The attempt at a solution

    I tried first by splitting the cosine up

    [tex] \int sin^2(x) [1-cos^2(x)] cos^2(x) cos(x) dx [/tex] and from there use u-substitution. However, I am not sure what to substitute. Any ideas?
    Last edited: Apr 13, 2014
  2. jcsd
  3. Apr 13, 2014 #2


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    $$ \int sin^2(\pi x) cos^5 (\pi x) dx $$
    $$ = \int sin^2(\pi x) cos^4 (\pi x) cos(\pi x) dx $$
    $$ = \int sin^2(\pi x) (1 - sin^2(\pi x))^2 cos(\pi x) dx $$

    Since the power of sin was even and cos was odd, you should save a factor of ##cos(x)## and convert the remaining ##cos^2(x)## terms to their ##1 - sin^2(x)## equivalents.

    Can you see a substitution from here that would help?
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