# Differentiation the integral

justinis123

## Homework Statement

Let f (x) =int(x,0) x sin(t^2)dt. Show that f''(x)= 2 sin(x^2) + 2x2 cos(x^2)

## The Attempt at a Solution

I cant get f''(x)= 2 sin(x^2) + 2x2 cos(x^2), i can only get f''(x)= sin(x^2) + 2x2 cos(x^2).
Because f'(x)=xsin(x^2). can anyone see the problem?

## Homework Statement

Let f (x) =int(x,0) x sin(t^2)dt. Show that f''(x)= 2 sin(x^2) + 2x^2 cos(x^2)

1. Since the integral is with respect to "t" then you can take the factor of "x" out the front of the integral.

2. You should now be able to use the product rule and the fundamental Thm of calculus to get the answer without ever actually evaluating the integral.

BTW. I assumed that by "2x2" you actually meant 2x^2".

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## The Attempt at a Solution

I cant get f''(x)= 2 sin(x^2) + 2x2 cos(x^2), i can only get f''(x)= sin(x^2) + 2x2 cos(x^2).
Because f'(x)=xsin(x^2). can anyone see the problem?

Yes that is the problem right there. See above about using the product rule to find f'.

justinis123
Hi uart
Thanks for the reply. Yeah, that was a typo. It should be 2x^2 as you assumed.
what do u mean by using product rule to find f' ?
f'(x)=xsin(x^2), then using product rule to find f''(x)=sin(x^2) + 2x^2 cos(x^2).
Could you please show me how to find f'(x)? I am a bit confused.
thanks