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The integral int(2x^(12)+5x^(9))/((x^(5)...

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))+C`

B

`(x^(5))/(2(x^(5)+x^(3)+1)^(2))+C`

C

`(-x^(10))/(2(x^(5)+x^(3)+1)^(2))`

D

`(-x^(5))/((x^(5)+x^(3)+1)^(2))+C`

Text Solution

Verified by Experts

The correct Answer is:
A
` I=int(2x^(12)+5x^(9))/((x^(5)+x^(3)+1)^(3))dx`
`=int(((2)/(x^(3))+(5)/(x^(6))))/((1+(1)/(x^(2))+(1)/(x^(5)))^(3))dx`
` "Let " 1+(1)/(x^(2))+(1)/(x^(5))=t`
` :. (dt)/(dx)=(-2)/(x^(3))-(5)/(x^(6))`
` :. int(-dt)/(t^(3))=(1)/(2t^(2))+C`
`=(1)/(2(1+(1)/(x^(2))+(1)/(x^(5)))^(2))+C`
`=(x^(10))/(2(x^(5)+x^(3)+1)^(2))+C`
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