Evaluate : `int x^n log x dx.`

Options

A. `(x)/(nm +1) [ log x - (1)/((n +1))] +C`
B. `
(x^(n+1))/(nm +1) [ log x - (1)/((n +1))] +C`
C. `(x^(n+1))/(n +1) [ log x - (1)/((n +1))] +C`
D. `(x^(n+1))/(nm ) [ log x - (1)/((n +1))] +C`

Answer

B

Explanation

`int x^n log x dx = int log x x^n dx = log x int x^n dx -int[(d)/(dx) (log x) int x^n dx] dx`
`= log x xxx^(n+1)/(n +1) - int (1)/(x) . (x^(n +1))/(n +1) dx = (x^(n +1) log x)/(n +1) - (1)/(n+1) int x^n dx`
` = (x^(n+1)log x)/(n +1) - (x^(n+1))/((n +1)^2) +C = (x^(n+1))/(nm +1) [ log x - (1)/((n +1))] +C`