Second derivative .

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. Im studying all my old exams for my final in 2 days and this is the last of the problems that I cant 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 thats 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. Im not necessarily looking for an answer (although that would definately help), moreso just looking for some direction. Thanks in advance.

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lanedance
Homework Helper

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

Last edited:

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

lanedance
Homework Helper

cool