Question

F'(x) for F(x)=int(ln(1+t^2)) from -x^2 to x^2?

Answer 1

You have,
\[F(x)=\int^{x^2}_{-x^2}\ln(1+t^2)dt=\int^{\alpha(x)}_{-\alpha(x)}f(t)dt\]
So
\[F'(x)=f(x^2)\alpha'(x)-f(-x^2)(-\alpha'(x))\]
Try it. It is an application of the well known theorem,
http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus

Answer 2

thank you, originally i ended up with all terms canceling but i now i have \[8x ^{3}\ln(1+x^{4})\]
i hope this is correct?

Answer 3

Nearly this. I would say,
\[F'(x)=4x\ln(1+x^4)\]
As
\[\alpha'(x)=2x\]

Answer 4

thank you so much!