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The integral int[2x^[12]+5x^9]/[x^5+x^3...

The integral `int[2x^[12]+5x^9]/[x^5+x^3+1]^3.dx` is equal to-

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`

Text Solution

Verified by Experts

The correct Answer is:
B
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`
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