`int(dx)/(x(x^n+1))` is equal to

Options

A. `(1)/(n)log((x^(n))/(x^(n)+1))+c`
B. `(1)/(n)log((x^(n)+1)/(x^(n)))+c`
C. `log((x^(n))/(x^(n)+1))+c`
D. non of these

Answer

A

Explanation

`I=int(dx)/(x(x^(n)+1))=int(x^(n-1))/(x^(n)(x^(n)+1))dx`
`"Putting " x^(n)=t " so that " nx^(n-1)dx=dt, i.e.,`
` "we get " x^(n-1)dx=(1)/(n)dt`
`I=int((1)/(n)dt)/(t(t+1))=(1)/(n)int((1)/(t)-(1)/(t+1))dt`
`=(1)/(n)(log t-log(t+1))+C`
`=(1)/(n)log((x^(n))/(x^(n)+1))+C`