Solution assignment 01 Integration by parts

Calculate:

$\displaystyle\int{\displaystyle\frac{1}{x}\ln(x)}dx$

We rewrite the integral as:

$\displaystyle\int{\ln(x).\displaystyle\frac{1}{x}}dx=\displaystyle\int{\ln(x)d(\ln(x))}$

because we have:

$\displaystyle\frac{d\ln(x)}{dx}=\displaystyle\frac{1}{x}$

and thus:

$d\ln(x)=\displaystyle\frac{1}{x}dx$

When we substitute in left-hand side:

$u=\ln(x)$

then we get:

$\displaystyle\int{u}du=\displaystyle\frac{1}{2}u^2+C=\displaystyle\frac{1}{2}\ln^2(x)+C$

We can verify this result by differentiation.