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The value of int((x^(2)+1))/(x^(4)+x^(2)...

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

A

`(1)/(sqrt3)tan^(-1){(x-1//x)/(sqrt3)}+C`

B

`(1)/(2sqrt3)log{((x-1//x)-sqrt3)/((x-1//x)+sqrt3))}+C`

C

`tan^(-1)((x+1//x)/(sqrt3)+C)`

D

`tan^(-1)((x-1/x)/(sqrt3))+C`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `l=int(x^(2)+1)/(x^(4)+x^(2)+1)dx=int((1+(1)/(x^(2))))/(x^(2)+1+(1)/(x^(2)))dx`
`=int((1+(1)/(x^(2))))/((x-(1)/(x))^(2)+(sqrt3)^(2))dx`
`=int(dt)/((sqrt3)^(2)+t^(2))=(1)/(sqrt3)tan^(-1)((t)/(sqrt3))+C`
`" "[" let "t=x-(1)/(x) rArr dt=(1+(1)/(x^(2)))dx]`
`=(1)/(sqrt3)tan^(-1){(1)/(sqrt3)(x-(1)/(x))}+C`
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