MHB Patrick's question at Yahoo Answers (First fundamental theorem of Calculus)

[Fernando Revilla]

Here is the question:

don't know how to type the question so here's the link to the image.

http://goo.gl/vQhhs

Thanks in advance :)

Here is a link to the question:

Integration by parts? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello Patrick,

The problem is:

Find $\dfrac{f''(2)}{\pi}$ if $f(x)=\displaystyle\int_1^x\sin (\pi t^2)\;dt$. Enter your answer as an integer.

Solution. Using the First fundamental theorem of Calculus, $f'(x)=\sin (\pi x^2)$. Deriving again, $f''(x)=2\pi x\cos (\pi x^2)$ so, $$\dfrac{f''(2)}{\pi}=\frac{4\pi\cos(4\pi)}{\pi}=4\cos(4\pi)=4$$