Answer
$\int x^\frac{3}{2}lnxdx = \frac{2x^{\frac{5}{2}}(5lnx - 2)}{25} + C$
Work Step by Step
The given indefinite integral is ,
$\int x^\frac{3}{2}lnxdx$
Solve this with integration by parts below :
The formula of integration by parts is :
$\int udv = uv - \int vdu $
Let's consider,
$u = lnx$
So, $du=\frac{1}{x}dx$
And,
$dv = x^{\frac{3}{2}}dx$
$v = \frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}$
So, $v = \frac{2}{5}x^{\frac{5}{2}}$
So, according to integration by parts,
$\int x^\frac{3}{2}lnxdx = \frac{2}{5}x^{\frac{5}{2}}lnx - \int \frac{2}{5}x^{\frac{5}{2}}\frac{1}{x}dx$
$\int x^\frac{3}{2}lnxdx = \frac{2}{5}x^{\frac{5}{2}}lnx -\frac{2}{5} \int \frac{x^{\frac{5}{2}}}{x}dx$
$\int x^\frac{3}{2}lnxdx = \frac{2}{5}x^{\frac{5}{2}}lnx -\frac{2}{5} \int x^{\frac{5}{2}-1}dx$
$\int x^\frac{3}{2}lnxdx = \frac{2}{5}x^{\frac{5}{2}}lnx -\frac{2}{5} \int x^{\frac{3}{2}}dx$
$\int x^\frac{3}{2}lnxdx = \frac{2}{5}x^{\frac{5}{2}}lnx -\frac{2}{5} \frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1} + C$
$\int x^\frac{3}{2}lnxdx = \frac{2}{5}x^{\frac{5}{2}}lnx -\frac{2}{5} \frac{x^{\frac{5}{2}}}{\frac{5}{2}} + C$
$\int x^\frac{3}{2}lnxdx = \frac{2}{5}x^{\frac{5}{2}}lnx -\frac{4}{25}x^{\frac{5}{2}} + C$
$\int x^\frac{3}{2}lnxdx = \frac{10x^{\frac{5}{2}}lnx - 4x^{\frac{5}{2}}}{25} + C$
$\int x^\frac{3}{2}lnxdx = \frac{2x^{\frac{5}{2}}(5lnx - 2)}{25} + C$
