Solution assignment 07 Integration by parts

Calculate:

$\displaystyle\int{x\ln(x)}dx$

We can write this integral as:

$\displaystyle\int{\ln(x)}d(\displaystyle\frac{1}{2}x^2)=$

$\displaystyle\frac{1}{2}\displaystyle\int{\ln(x)}d(x^2)=$

$\displaystyle\frac{1}{2}[x^2\ln(x)-\displaystyle\int{x^2}d(\ln(x)]$

$\displaystyle\frac{1}{2}[x^2\ln(x)-\displaystyle\int{x^2}\displaystyle\frac{1}{x}dx]$

$=\displaystyle\frac{1}{2}[x^2\ln(x)-\displaystyle\int{x}dx]=$

$=\displaystyle\frac{1}{2}[x^2\ln(x)-\displaystyle\frac{1}{2}x^2]+C$

Verify the result by differentiation.