Originally Posted by

**HallsofIvy** There is no need to actually do the integral- this is an exercise in using the "Fundamental Theorem of Calculus":

$\displaystyle \frac{d}{dx}\int_a^x f(t)dt= f(x)$

Here, because the upper limit is $\displaystyle x^2$ rather than x, you need to use the substitution $\displaystyle u= \sqrt{t}= t^{1/2}$. That way the upper limit is $\displaystyle \sqrt{x^2}= x$ (assuming x is positive). Now $\displaystyle du= (1/2)t^{1/2}dt$ and $\displaystyle t= u^2$ so that $\displaystyle dt= \frac{2du}{t^{1/2}}= 2\frac{du}{u}$ and the integral becomes

$\displaystyle 2\int_{\sqrt{3}}^x (2u+ 5)^2\left(\frac{du}{u}\right)$

and, by the Fundamental Theorem of Mathematics, the derivative of that is

$\displaystyle 2\frac{(2x+ 5)^2}{x}$

just by replacing u in the integrand by x.