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Thread: substitution problem

  1. #1
    Member Em Yeu Anh's Avatar
    Joined
    Nov 2009
    Posts
    94

    Angry substitution problem

    If f is continuous on [0,$\displaystyle \pi$], use the substitution $\displaystyle u = \pi - x$ to prove that:

    $\displaystyle \int_0^{\pi}xf(sinx)dx = \frac{\pi}{2}\int_0^{\pi}f(sinx)dx$

    Completely lost here. $\displaystyle du = -1dx $ and I don't know how to start from there.
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  2. #2
    Senior Member
    Joined
    Dec 2008
    Posts
    319
    From

    $\displaystyle \begin{aligned}
    u&=\pi-x\\
    du&=-dx
    \end{aligned}$

    we may derive

    $\displaystyle \begin{aligned}
    x&=\pi-u\\
    dx&=-du.
    \end{aligned}$

    Hope this helps!
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