Integrating Trigonometric Functions and Evaluating Second Derivatives

[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)

Homework Equations

The Attempt at a Solution

I can't 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?

 

[uart]

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".

 

[uart]

The Attempt at a Solution

I can't 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

 

[uart]

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

You're making your mistake at the first step in finding f'. You need to use the product rule to find f' (and then again to find f''). Re-read my first post about taking the "x" out the front of the original integral before you even start, this is the key.

BTW. The expression for f' that you obtained would be correct if the integrand was t sin(t^2) dt but it' not. It's x sin(t^2)dt. So you have to re think your original approach and follow my advice.

 

[justinis123]

Thanks , i see the problem. I fixed the issue and got the right answer.