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

MHB Math Helper

#### Fernando Revilla

##### Well-known member
MHB Math Helper
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, $$\dfrac{f''(2)}{\pi}=\frac{4\pi\cos(4\pi)}{\pi}=4\cos(4\pi)=4$$