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Math Help - Evaluate integral?

  1. #1
    Super Member fardeen_gen's Avatar
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    Evaluate integral?

    Evaluate:
    \int_{0}^{\pi} e^{|\cos x|} \left\{2\sin\left(\frac{1}{2}\cos x\right) + 3\cos\left(\frac{1}{2}\cos x\right)\right\}\sin x dx
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  2. #2
    Super Member Gamma's Avatar
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    I think you want to break the integral into two parts by the sum. Then make the u substitution u=1/2cos(x). That will get you the sin(x)dx you need. so you will have a -2e^{|2u|}(a)sin/cos(u)du
    May need to break this integral up again to deal with that absolute value.

    Then I think just do the integration by parts twice trick and you will get back something familar and solve that way. I assume you know what you are doing if you were assigned this kind of problem, so I left out a lot of steps. Let me know if you need me to fill in details.
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  3. #3
    Super Member Gamma's Avatar
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    ooh, i just started working it out a bit and I got the first one to reduce to

    4\int_{-1/2}^{1/2}e^{|2u|}sin(u)du

    But this is an odd function, with symmetric limits of integration so it should be 0.


    For the second one
    I get 6\int_{-1/2}^{1/2}e^{|2u|}cos(u)du which is an even function, so this is going to be
    12\int_{0}^{1/2}e^{2u}cos(u)du

    which yeah should be able to just do the integration by parts twice to solve.

    Be advised I did these on the back of an envelope really quicky, but hope that helps.
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