Question

Integrate the function: $x log \ 2x$

Answer 1

Let I=∫x log 2x dx

Taking log 2x as first function and x as second function and integrating by parts, we obtain

I = log 2x∫x dx-∫{($\frac {d}{dx} 2log \ x$)∫x dx} dx

I = log 2x . $\frac {x^2}{2}$ - ∫$\frac {2}{2x}$ . $\frac {x^2}{2}$dx

I = $\frac {x^2log\ 2x}{2}$ - ∫$\frac {x}{2}$dx

I = $\frac {x^2log\ 2x}{2}$ - $\frac {x^2}{4}$ + C