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November 29th, 2015, 06:58 PM  #1 
Newbie Joined: Nov 2015 From: Philippines Posts: 1 Thanks: 0  Application of First Fundamental Theorem of Calculus
Find the second derivative of f($x$)=$\displaystyle \int_{\infty}^{x^2/2} e^{(x^2+1)t^2} dt$ My solution: f($x$)=$\displaystyle \int_{\infty}^0 e^{(x^2+1)t^2}dt + \int_0^{x^2/2} e^{(x^2+1)t^2}dt$ I may be able to find a closed form for the first integral $\displaystyle \int_{\infty}^0 e^{(x^2+1)t^2}dt$ by transforming into polar coordinates. My concern is the second integral: h($x$) = $\displaystyle \int_0^{x^2/2} e^{(x^2+1)t^2}dt$ How do I apply Chain rule to get h'($x$)? Last edited by skipjack; November 29th, 2015 at 11:59 PM. Reason: to use latex 
November 30th, 2015, 12:05 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,723 Thanks: 1807 
I doubt that the final answer can be given in closed form. I suggest that you make a substitution so that the original integral can be written in terms of the error function, erf(x).

November 30th, 2015, 02:00 PM  #3 
Global Moderator Joined: May 2007 Posts: 6,613 Thanks: 617 
There is no point in splitting the integral. If you apply the basic definition of the derivative, $\displaystyle f'(x)=\lim_{h >0}\frac{f(x+h)f(x)}{h}$, you will be able to work it out by brute force. For your problem let $\displaystyle f(x)=\int_{\infty}^{g(x)}p(x,t)dt$ $\displaystyle f'(x)=p(x,g(x))g'(x)+\int_{\infty}^{g(x)}\frac{\partial p(x,t)}{\partial x}dt$ I'll let you work out the second derivative. 

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