Beginner Question on Integral Calculus

I have two questions attached, the fact that the function is f(x) but the variables are t is throwing me off.

Also, does the derivative cancel out the integral and just leave the function as is?

I appreciate any solution & quick explanation.

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  • So you can just replace the t's with x's? Could you also explain how you got from the 3rd line to the 4th line in Q2?

  • Yes. In Q2 going from 3rd line to the 4th line, note that the derivative of 2pi x^2 is 4pi x.

  • Also sin (32 pi)=0.

The answer is accepted.