# How do you integrate int x^(1/2)*ln(x)  using integration by parts?

Nov 29, 2015

$\frac{2}{9} {x}^{\frac{3}{2}} \left(3 \ln \left(x\right) - 2\right) + C$

#### Explanation:

Integration by part :
$\int f \left(x\right) g ' \left(x\right) \mathrm{dx} = f \left(x\right) g \left(x\right) - \int f ' \left(x\right) g \left(x\right) \mathrm{dx}$

Given $\int \sqrt[]{x} \cdot \ln \left(x\right) \mathrm{dx}$

Let $f \left(x\right) = \ln x$
color(blue)(f'(x)= 1/x dx $\text{ " " " " } \left(1\right)$

Let g'(x)= x^(1/2)dx hArr g(x) = int x^(1/2)dx =>color(red)( g(x)= 2/3 x^(3/2) $\text{ " " " " } \left(2\right)$

$\int \sqrt[]{x} \cdot \ln \left(x\right) \mathrm{dx}$ = color(red)(2/3 x^(3/2))*ln(x)- intcolor(red)( " " 2/3 x^(3/2)*color(blue)((1/x)dx)

$\frac{2}{3} {x}^{\frac{3}{2}} \ln \left(x\right) - \frac{2}{3} \int {x}^{\frac{1}{2}} \mathrm{dx}$

$\frac{2}{3} {x}^{\frac{3}{2}} \ln \left(x\right) - \frac{2}{3} \cdot \frac{2}{3} \left({x}^{\frac{3}{2}}\right) + C$

$\frac{2}{3} {x}^{\frac{3}{2}} \ln \left(x\right) - \frac{4}{9} \left({x}^{\frac{3}{2}}\right) + C$

Can be simplify to
$\frac{2}{9} {x}^{\frac{3}{2}} \left(3 \ln \left(x\right) - 2\right) + C$