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Math Help - Trig substitution

  1. #1
    Junior Member Hardwork's Avatar
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    Trig substitution

    How can we find the integral \int_{0}^{\pi}\cos{x}\sin{x}\,{dx} with the substitution u = sinx?

    I know how to find the integral, but the limits become the same sin(0) = sin(pi) = 0.

    What should I do in general when a given substitution makes upper bound = lower bound?
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  2. #2
    Super Member TheChaz's Avatar
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    Quote Originally Posted by Hardwork View Post
    How can we find the integral \int_{0}^{\pi}\cos{x}\sin{x}\,{dx} with the substitution u = sinx?

    I know how to find the integral, but the limits become the same sin(0) = sin(pi) = 0.

    What should I do in general when a given substitution makes upper bound = lower bound?
    Using a different property, you can analyse the integrand as a function with period "pi", the (signed) area of which equals zero after each period.
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  3. #3
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by Hardwork View Post
    How can we find the integral \int_{0}^{\pi}\cos{x}\sin{x}\,{dx} with the substitution u = sinx?

    I know how to find the integral, but the limits become the same sin(0) = sin(pi) = 0.

    What should I do in general when a given substitution makes upper bound = lower bound?
    Is...





    ... so that, no matter which 'substitution' You do, the integral is 0...

    Kind regards

    \chi \sigma
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  4. #4
    Junior Member Hardwork's Avatar
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    Okay. I do see how the integral is zero. I just want to know what to do in general when a substitution makes the limits the same.
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