The integral `int[2x^[12]+5x^9]/[x^5+x^3+1]^3.dx` is equal to- (A) `x^10 / (2(x^5 + x^3 +1)^2) ` (B) `x^5/ (2(x^5 + x^3 +1)^2) ` (C) `-x^10 / (2(x^5 + x^3 +1)^2) ` (D) `- x^5 / (2(x^5 + x^3 +1)^2) `

Options

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

Answer

B

Explanation

Let `I=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`
Now, put `1 + x^(-2)+ x^(-5) = t`
`rArr (-2x^(-3)-5x^(-6))dx=dt`
`rArr (2x^(-3)+5x^(-6))dx=dt`
`therefore I= - int(dt)/(t^(3))=-int t^(-3)dt`
`=-(t^(-3+1))/(-3+1)+C=(1)/(2t^(2))+C`
`=(x^(10))/(2(x^(5)+x^(3)+1)^(2))+C`