Question

Evaluate the integral using integration by parts with the indicated choices of u and dv. (Use C for the constant of integration.) ∫4x2 lnx dx ; u= lnx , dv=4x 2dx

Answer 1

Take

$u=\ln x\implies\mathrm du=\dfrac{\mathrm dx}x$

$\mathrm dv=4x^2\,\mathrm dx\implies v=\dfrac43x^3$

Then

$\displaystyle\int4x^2\ln x\,\mathrm dx=\frac43x^3\ln x-\frac43\int x^2\,\mathrm dx=\frac43x^3\ln x-\frac49x^3+C$

$=\boxed{\dfrac49x^3(3\ln x-1)+C}$