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int(2x^(2)(1+2x^(2)))/((1+x^(2)+x^(4))^(...

`int(2x^(2)(1+2x^(2)))/((1+x^(2)+x^(4))^(2))dx` is equal to

A

`-(x)/(1+x^(2)+x^(4))+(1)/(2sqrt(3))tan^(-1)((x-(1)/(x))/(sqrt(3)))-(1)/(4)log""((x+(1)/(x)-1)/(x+(1)/(x)+1))+C`

B

`-(x)/(1+x^(2)+x^(4))+(1)/(2sqrt(3))tan^(-1)((x+(1)/(x))/(sqrt(3)))+(1)/(2)-log""((x+(1)/(x)-1)/(x+(1)/(x)+1))+C`

C

`-(x)/(1+x^(2)+x^(4))+(1)/(2sqrt(3))tan^(-1)((x-(1)/(x))/(sqrt(3)))+(1)/(4)+ log""((x+(1)/(x)-1)/(x+(1)/(x)-1))+C`

D

`-(x)/(1+x^(2)+x^(4))+(1)/(2sqrt(3))tan^(-1)((x-(1)/(x))/(sqrt(3)))+(1)/(4)log((x+(1)/(x)-4)/(x+(1)/(x)-4))+C`

Text Solution

Verified by Experts

The correct Answer is:
A
`intx((2x+4x^(3)))/((1+x^(2)+x^(4)))dx`
`= x(-(1)/(1+x^(2)+x^(4)))-int-(1)/(1+x^(2)+x^(4))dx`
`=-(x)/(1+x^(2)+x^(4))+(1)/(2) int(1+x^(2)+1-x^(2))/(1+x^(2)+x^(4))dx`
`=-(x)/(1+x^(2)+x^(4))+(1)/(2)int(1+x^(2))/(1+x^(2)+x^(4))dx+(1)/(2)int(1-x^(2))/(1+x^(2)+x^(4))dx`
`=-(x)/(1+x^(2)+x^(4))+(1)/(2)int(1+(1)/(x^(2)))/((x-(1)/(x))^(2)+3)dx+(1)/(2)int((-1)(1-(1)/(x^(2))))/((x+1(1)/(x))^(2)-1)dx`
`=-(x)/(1+x^(2)+x^(4))+(1)/(2sqrt(3))tan^(-1)((x-(1)/(x))/(sqrt(3)))-(1)/(4)log""((x+(1)/(x)-1)/(x+(1)/(x)+1))+C`
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