Question

fundamental theorem or calculus help? (problem in comments.)

Answer 1

\[\frac{ d }{ dx }(\int\limits_{0}^{x^{4}}\ln (t^{2}+1)dt)\]

Answer 2

@ganeshie8

Answer 3

@agent0smith

Answer 4

\(\frac{ d }{ dx }(\int\limits_{0}^{x^{4}}\ln (t^{2}+1)dt) \\ = \dfrac{d}{dx}\left[F(x^4) - F(0)\right] \\= F'(x^4) (x^4)' - 0 \\ = \ln((x^4)^2 + 1) * 4x^3\)

Answer 5

btw, your \(F'(t) = \ln(t^2+1)\) \(\implies F(t) = \int F'(t) dt \)

Answer 6

Ah, I see! thanks for your help.

Answer 7

In short :
just apply the FTC, and apply chain rule for the function in bounds

Answer 8

\(\frac{ d }{ dx }(\int\limits_{0}^{x^{4}}\ln (t^{2}+1)dt) = \ln((x^4)^2 + 1) * (x^4)' \)