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int(3x^(13)+2x^(11))/((2x^4+3x^2+1)^4)dx...

`int(3x^(13)+2x^(11))/((2x^4+3x^2+1)^4)dx`

A

`(x^(4))/(6(2x^(4)+3x^(2)+1)^(3))+C`

B

`(x^(12))/(6(2x^(4)+3x^(2)+1)^(3))+C`

C

`(x^(4))/((2x^(4)+3x^(2)+1)^(3))+C`

D

`(x^(12))/((2x^(4)+3x^(2)+1)^(3))+C`

Text Solution

Verified by Experts

The correct Answer is:
B
Let `I=int(3x^(13)+2x^(11))/((2x^(4)+3x^(2)+1)^(4))dx = int((3)/(x^(3))+(2)/(x^(5)))/((2+(3)/(x^(2))+(1)/(x^(4)))^(4))dx` [on dividing numerator and denominator by `x^(16)`]
Now, put `2 + (3)/(x^(2))+(1)/(x^(4))=t`
`rArr ((-6)/(x^(3))-(4)/(x^(5)))dx = dt rArr ((3)/(x^(3))+(2)/(x^(5)))dx = - (dt)/(2)`
So, `I = int(-dt)/(2t^(4))= -(1)/(2)xx(t^(-4+1))/(-4+1)+C = (1)/(6t^(3))+C`
`= (1)/(6(2+(3)/(x^(2))+(1)/(x^(4)))^(3))+C" "[therefore t = 2 + (3)/(x^(2))+(1)/(x^(4))]`
`= (x^(12))/(6(2x^(4)+3x^(2)+1)^(3))+C`
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