Second derivative of a nested integral with Leibniz rule

[mateomy]

Second derivative...

Homework Statement

Okay, this is a rough one for me. It was a question I got on my test, and (obviouslly) didnt get right. I am studying all my old exams for my final in 2 days and this is the last of the problems that I can't wrap my head around...any help would be greatly appreciated.

$$\frac{d^2}{dx^2} \int_0^{x}\,(\Big\,\int_1^{sint}\sqrt{1+u^2}\,du)\Big\,dt$$

I know, through the fundamental theorem of calculus that I can just replace (so to speak) the 't' in 'sint' to a 'sinx'. and then replace the $u^2$ with 'sinx'. I think that's it for the first part, then I think to find the second derivative I just derive it again? I dunno...Im seriously lost; all those parenthesis and variable changes are throwing me off. I am not necessarily looking for an answer (although that would definitely help), more so just looking for some direction. Thanks in advance.

 

[lanedance]

reading through this should help http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign

so start by differentiating once, and only term from the x in the limit will contribute

if it helps consider it as:
$$I(x) = \int_0^{x} g(t) dt$$

where
$$g(t) = \int_1^{sint}\sqrt{1+u^2} du$$

then you will need to think about the 2nd differentiation step

 

[mateomy]

Sweet. Got it, solved it, psyched on it. Thank you very much for the direction.

 

[lanedance]

cool