Question

What is the Integration of 2x³Inx​

Answer 1

`~~~\displaystyle \int 2x^3 \ln x ~ dx\\\\\\=2\displaystyle \int x^3 \ln x ~dx\\\\\\=2 \left[\ln x \displaystyle \int x^3 ~ dx - \displaystyle \int \left( (d)/(dx) \ln x \displaystyle \int x^3~ dx \right) ~ dx\right]~~~~~~~~~~~~~~~~~~;\text{Integration by parts}\\\\\\=2 \left[ \frac{x^4 \ln x}4 - \displaystyle \int \left( \frac 1x \cdot (x^4)/(4) \right)~ dx \right]\\\\\\=2\left[ (x^4 \ln x)/(4) - \frac 14 \displaystyle \int x^3 ~ dx \right]\\\\\\`

`=2 \left( (x^4 \ln x )/(4) - \frac 14 \cdot (x^4 )/(4 ) \right) +C\\\\\\=(x^4 \ln x )/(2) - (x^4 )/(8) +C\\\\\\=(4x^4 \ln x - x^4 )/(8) +C\\\\\\=\frac 18 x^4\left(4 \ln x -1\right) +C`