Question

True or False

To evaluate the integral $\displaystyle{\int \sqrt{x}\ln{(x)}dx}$  by integration by parts,  it is wise to choose $u=\sqrt{x}$ and $dv=\ln{(x)}dx$

Answer 1

False

 it is not wise to choose $u=\sqrt{x}$ and $dv=\ln{(x)}dx$ .

Since the anti derivative of $\sqrt{x}$ , is known but the anti derivative of $lnx$ is not. So $u=ln(x)$ and $dv=\sqrt{x}dx$ ​

$du=\frac{1}{x}$

$v=\frac{2x^{\frac{3}{2}}}{3}$

This because the derivative of $ln(x)$ which is $\frac{1}{x}$ , will interact nicely with polynomial terms, whereas the integral is some weird derivative on the order of $xlnx$ which looks terrible