`int (2x^(12) +5x^(9))/((x^(5)+x^(3)+1)^(3)) dx` बराबर है:

Options

A. `(-x^(5))/((x^(5)+x^(3)+1)^(2))+c`
B. `x^(10)/(2(x^(5)+x^(3)+1)^(3))+c`
C. `x^(5)/(2(x^(5)+x^(3)+1)^(2))+c`
D. `(x^(10))/(2(x^(5)+x^(3)+1)^(2))+c`.

Answer

D

Explanation

`int (2x^(12)+5x^(9))/((x^(5)+x^(3)+1)^(3)) dx=int (2x^(12)+5x^(9))/(x^(15) (1+x^(-2)+x^(-5))^(3)) dx`
`=int (2x^(-3)+5x^(-6))/((1+x^(-2)+x^(-5))^(3)) dx`
मान लीजिए `1+x^(-2)+x^(-5)=t`
`implies (-2x^(-3)-5x^(-6)) dx=dt`
`implies (2x^(-3)+5x^(-6)) dx=-dt`
`=-int (dt)/t^(3)`
`=-int t^(-3) dt`
`=- t^(-3+1)/(-3+1) +c`
`=1/(2t^(2))+c`
`=1/(2(1+x^(-2)+x^(-5))^(2))+c`
`=1/(2x^(-10) (x^(5)+x^(3)+1)^(2))+c`
`= x^(10)/(2(x^(5)+x^(3)+1)^(2))+c`