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Math Help - Derivative of an Integral Question

  1. #1
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    Derivative of an Integral Question

    Find  \frac {d^2}{dx^2} \int^x_0(\int^{sint}_1 \sqrt{1+u^4}du)dt

    So these were my steps:

     \frac {d}{dx} (\frac {d}{dx}\int^x_0(\int^{sint}_1 \sqrt{1+u^4}du)dt)


     \frac {d}{dx} (\int^{sint}_1 \sqrt{1+x^4}dt)

    I am not sure if i followed the FTC correctly with the inside d/dx and integral by changing u to x, if I did I think I managed to follow the rest through correctly the prebious part was what I was confused on.

     \sqrt{1+sint^4}(sint) dt

     \sqrt{1+sint^4}(cost)

    I think I was suppose to start from the inside and go out but I did not know anyway to integrate \sqrt{1+u^4}
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    Put f(x)=\int_0^x g(t)\,dt, where g(t)=\int_1^{\sin t}\sqrt{1+u^4}\,du, then f'(x)=\int_1^{\sin x}\sqrt{1+u^4}\,du so f''(x) ?
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