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Trig substitution integration

  1. Jan 24, 2017 #1
    1. The problem statement, all variables and given/known data

    ∫8cos^3(2θ)sin(2θ)dθ
    2. Relevant equations


    3. The attempt at a solution
    rewrote the integral as:
    8∫(1-sin^2(2θ))sin(2θ)cos(2θ)dθ
    u substitution with u=sin(2θ) du=2cos(2θ)dθ
    4∫(1-u^2)u du= 4∫u-u^3 du
    4(u^2/2-u^4/4)+C
    undo substitution and simplify
    2sin^2(2θ)-sin^4(2θ)+C
    The book gives an answer of:
    -cos^4(2θ)+C
     
  2. jcsd
  3. Jan 24, 2017 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Check your answer by differentiating it to see if you get back to ##8 \cos^3 (2 \theta) \sin (2 \theta)##. Checking your answer by differentiation is something you should always do, every time.
     
  4. Jan 24, 2017 #3
    I do in fact get my original expression. Thanks.
     
  5. Jan 24, 2017 #4

    Mark44

    Staff: Mentor

    There's no need to rewrite the integral. In the original integral, use ordinary substitution with ##u = \cos(2\theta)##, and ##du = -2\sin(2\theta)d\theta##. Using this substitution leads directly to the book's answer.
    I second Ray's advice to always check your answer by differentiation.
     
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