Question

Evaluate: $\int x\cdotp\log {x} \, dx$

Answer 1

Solution:

To evaluate the integral ,$\int x\cdotp\log {x} \, dx$ we have to apply formula of integration by parts

$\int {P .Q} dx = P \int Q dx-\int [\frac{dP}{dx}\int Q dx]dx$

Here we assume P = log x

Q= x

$\int x.logx\:dx = logx\int x\:dx-\int[\frac{d logx}{dx}\int\:x dx]dx\\\\\\=\frac{logx.{x}^{2}}{2}-\int[\frac{1}{x}]\frac{{x}^{2}}{2}dx\\\\\\=\frac{logx.x^{2}}{2}-\frac{{x}^{2}}{2}+C$

Is the final solution