# Math Help - Derivative of an Integral Question

1. ## 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 $\sqrt{1+u^4}$

2. 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 $f''(x)$ ?