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Math Help - Help with showing integral is differentiable

  1. #1
    owq
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    Question Help with showing integral is differentiable

    f is continuous on x=[0,1].

    Given that \int_0^{\pi}\\xf(sin x)\ dx = \frac{\pi}{2}\int_0^{\pi}\\f(sinx) dx

    Show that
    F(x)=\int_0^1\\f(x+t) dt is differentiable and that F\prime(x)=f(x+1)-f(x).
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  2. #2
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    i think the first relation has nothing to do here.

    put u=x+t in your integral and then F(x)=\int_x^{x+1}f(u)\,du, and f was given as continuous, so F is differentiable and then F'(x)=f(x+1)-f(x).
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