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