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Thread: Compute the integral using trig substitution

  1. #1

    Compute the integral using trig substitution

    I need to correct the following problem using trig substitution.
    eqn4315.png


    I keep getting stuck at:

    eqn4315.png

    If I dont use trig sub and just use u-sub I get like 2/15, but I think I need to answer using trig sub. Thanks for any help you can offer!

  2. #2
    Elite Member
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    Quote Originally Posted by imattxc View Post
    I need to correct the following problem using trig substitution.
    eqn4315.png


    I keep getting stuck at:

    eqn4315.png

    If I dont use trig sub and just use u-sub I get like 2/15, but I think I need to answer using trig sub. Thanks for any help you can offer!


    sin3(Θ) * cos2(Θ) = sin(Θ) * [1-cos2(Θ)] * cos2(Θ) = sin(Θ) * cos2(Θ) - sin(Θ) * cos4(Θ) .... Now integrate....
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  3. #3
    Elite Member
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    Typically, if you have an odd power of sine or cosine you can factor one out, to use with the "dx", then use [tex]sin^2(x)+ cos^2(x) = 1[/tex], so that either [tex]sin^2(x)= 1- cos^2(x)[/tex] or [tex]cos^2(x)= 1- sin^2(x)[/tex] to reduce the remaining even power. Here, you have [tex]\int_0^{\pi/2} sin^3(\theta) cos^2(\theta)d\theta[/tex] which we can write as [tex]\int_0^{\pi/2} sin^2(\theta) cos^2(\theta) (sin(\theta) d\theta)= \int_0^{\pi/2} (1- cos^2(\theta))cos^2(\theta)(sin(\theta)d\theta)= \int_0^{\pi/2} (cos^2(\theta)- cos^4(\theta))(sin(\theta)d\theta)[/tex].

    Now make the substitution [tex]u= cos(\theta)[/tex].
    Last edited by Subhotosh Khan; 11-17-2014 at 11:35 AM. Reason: typo

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