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

AI Thread Summary
The discussion revolves around a calculus problem involving the First Fundamental Theorem of Calculus. The question asks for the value of f''(2) divided by π, where f(x) is defined as the integral of sin(πt²) from 1 to x. The solution involves differentiating the integral to find f'(x) and then f''(x), leading to the calculation of f''(2). The final answer is determined to be 4 after simplifying the expression. This highlights the application of calculus principles in solving integral problems.
Fernando Revilla
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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.
 
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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, $$\dfrac{f''(2)}{\pi}=\frac{4\pi\cos(4\pi)}{\pi}=4\cos(4\pi)=4$$
 
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