If m is a non-zero number and `int (x^(5m-1)+2x^(4m-1))/(x^(2m)+x^m+1)^3 dx=f(x)+c`, then f(x) is:

Options

A. `( x^(6m)+x^(2m))/(2m(x^(2m)+x^(m)+1)^(2))+c`
B. `(x^(4m))/(2m(x^(2m)+x^(m)+1)^(2))+c`
C. `(2m x^(4m))/((x^(2m)+x^(m)+1)^(2))+c`
D. `(m x^(5m))/(2(x^(2m)+x^(m)+1)^(2))+c`

Answer

B

Explanation

`int(x^(5m-1)+2x^(4m-1))/(x^(6m)(1+x^(-m)+x^(-2m))^(3))dx=int(x^(-(m+1))+2*x^(-(2m+1)))/((1+x^(-m)+x^(-2m))^(3))dx`
Put `(1+x^(-m)+x^(-2m))=t`
` :. (-1)/(m)int(dt)/(t^(3))=(1)/(2mt^(2))+c=(x^(4m))/(2m(x^(2m)+x^(m)+1)^(2))+c`