Solution assignment 06 Integration by parts

Calculate:

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

We do not like the root as argument of the $\ln$ -function and thus we first apply the substitution method:

$u=\sqrt{x}$

or:

$u^2=x$

Differentiation of this last formule yields:

$2udu=dx$

Now we can rewrite the integral as follows:

$\displaystyle\int{\ln(\sqrt{x})}dx=2\displaystyle\int{u\ln(u)}du$

At this point we can apply integration by parts:

$2\displaystyle\int{u\ln(u)}du=$

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

$=u^2\ln(u)-\displaystyle\int{u^2}d(\ln(u))=$

$=u^2\ln(u)-\displaystyle\int{u}du=$

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

Verify the result by differentiation.