Generalisation of Fundamental Theorem of Calculus

Question

How does the theorem hold if the limits to the integral aren't $0$ and $x$? Say if it was bounded by $x^2$ and $0$. Or does it only apply when the limits are the ones given in the definition?

Answer 1

If $f$ is continuous and $F(x)=\int_0^{x^2}f(t)\,\mathrm dt$, then $F=G\circ q$, where $G(x)=\int_0^xf(t)\,\mathrm dt$ and $q(x)=x^2$. Therefore, you can apply the chain rule:$$F'(x)=G'\bigl(q(x)\bigr)q'(x)=f(x^2)\times(2x).$$