Question

Find the derivative with respect to x of the integral from 2 to x^3 of ln(x^2)dx

Answer 1

The derivative follows from the fundamental theorem of calculus:

$\displaystyle\frac{\mathrm d}{\mathrm dx}\int_c^{g(x)}f(t)\,\mathrm dt=f(g(x))\dfrac{\mathrm dg}{\mathrm dx}$

where $c$ is any constant in the domain of $f$.

We have

$g(x)=x^3\implies\dfrac{\mathrm dg}{\mathrm dx}=3x^2$

so

$\displaystyle\frac{\mathrm d}{\mathrm dx}\int_2^{x^3}\ln(t^2)\,\mathrm dt=3x^2\ln((x^3)^2)=18x^2\ln x$

(applying the property $\ln a^b=b\ln a$)