Question

Calcule a integral ∫ 1 0 x l n ( x ) d x

Answer 1

$u = lnx \\ du = \frac{1}{x} dx \\ dv = 10xdx \\ v = \int10xdx \\ v = 5 {x}^{2}$

$\int u.dv = u.v - \int \: v.du$

$\int 10xln(x)dx \\ = ln(x). {5x}^{2} -\int5 {x}^{2}. \frac{1}{x}dx \\ = ln(x).5 {x}^{2} - \int5xdx$

$\int10x ln(x)dx \\ = ln(x).5 {x}^{2} - \frac{5}{2} {x}^{2} + c$