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int (1+2x^(6))/((1-x^(6))^(3//2))dx is e...

`int (1+2x^(6))/((1-x^(6))^(3//2))dx` is equal to

A

`(2x)/(sqrt(1+x^(6)))+c`

B

`(1)/(sqrt((1)/(x^(2))-x^(4)))+c`

C

`(x)/(sqrt((1)/(x^(2))-x^(4)))+c`

D

`(x^(2)+x)/(sqrt(1-x^(6)))+C`

Text Solution

Verified by Experts

The correct Answer is:
B
`I=int (1+2x^(6))/((1-x^(6))^(3//2))dx=int((1)/(x^(3))+2x^(3))/((1)/(x^(2))-x^(4))^((3)/(2))dx`
put `(1)/(x^(2))-x^(4)=t`
` :. ((-2)/(x^(3))-4x^(3))dx=dt`
or `((1)/(x^(3))+2x^(3))dx= -(dt)/(2)`
` :. I= -(1)/(2)int (dt)/(t^((3)/(2)))`
`= -(1)/(2)(t^(-1//2))/(2-1//2)+c`
` =(1)/(sqrt(t))+c`
`=(1)/(sqrt((1)/(x^(2))-x^(4)))+c`
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