`int32x^(3)(logx)^(2)` dx is equal ot

Options

A. `8x^(4)(logx)^(2)+C`
B. `x^(4){8(logx)^(2)-4log x+1}+C`
C. `x^(4){8(logx)^(2)-4logx}+C`
D. `x^(3){(logx)^(2)+2logx}+C`

Answer

B

Explanation

`int 32x^(3)(logx)^(2)dx`
On using integration by parts, we get
`32{(logx)^(2).(x^(4))/(4)-int 2 log x.(1)/(x).(x^(4))/(4)dx}`
`=8x^(4)(logx)^(2)-16intx^(3)log x dx`
Again using integration by parts, we get
`8x^(4)(logx)^(2)-16{logx.(x^(4))/(4)-int(1)/(x).(x^(4))/(4)dx}`
`=8x^(4)(logx)^(2)-4x^(4)logx+4int x^(3)dx`
`=8x^(4)(logx)^(2)-4x^(4)logx+x^(4)+C`
`=x^(4)[8(logx)^(2)-4logx+1]+C`