 # Evaluate the integral. int x^2 log(4x) dx Adrianna Macias 2022-07-19 Answered
Evaluate the integral. $\int {x}^{2}\mathrm{log}\left(4x\right)dx$
The problem is $\int {x}^{2}\mathrm{log}\left(4x\right)dx$
Here $\mathrm{ln}$ refers to the natural logarithm.
So far, I know $u={x}^{2}$ and $du=2x\left(dx\right)$
So $dv=\mathrm{ln}\left(4x\right)dx$ and $v=1/x$, but I don't know where to go from here.
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Integrating by parts,
$\int {x}^{2}\mathrm{log}4xdx=\frac{1}{3}{x}^{3}\mathrm{log}4x-\int \frac{1}{3}{x}^{3}\cdot \frac{4}{4x}dx\phantom{\rule{0ex}{0ex}}=\frac{1}{3}{x}^{3}\mathrm{log}4x-\frac{1}{3}\int {x}^{2}dx\phantom{\rule{0ex}{0ex}}=\frac{1}{3}{x}^{3}\mathrm{log}4x-\frac{1}{9}{x}^{3}+C$

We have step-by-step solutions for your answer! Levi Rasmussen
$\int {x}^{2}\mathrm{ln}\left(4x\right)dx=\frac{1}{3}\int \mathrm{ln}\left(4x\right)d\left({x}^{3}\right)=\frac{1}{3}\left({x}^{3}\mathrm{ln}\left(4x\right)-\int {x}^{3}d\left(\mathrm{ln}\left(4x\right)\right)\right)$
$\int {x}^{3}d\left(\mathrm{ln}\left(4x\right)\right)=\int 4\frac{{x}^{3}}{4x}dx$
Thus simplifying your problem considerably.

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