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int(2)/((2-x)^(2))root3((2-x)/(2+x))dx i...

`int(2)/((2-x)^(2))root3((2-x)/(2+x))dx` is equal to

A

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

B

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

C

`(3)/(4)((2-x)/(2+x))^(2//3)`

D

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

Text Solution

Verified by Experts

The correct Answer is:
B
Put `(2-x)/(2+x)=z^(3) rArr x=(2-2z^(3))/(1+z^(3))`
`rArr" "dx=(-12z^(2)dz)/((1+z^(3))^(2))`
`therefore" "int(2)/((2-x)^(2)) root3((2-x)/(2+x))dx`
`=int(2)/([2-(2-2z^(3))/(1+z^(3))]^(2)).z.((-12z^(3)))/((1+z^(3))^(2))dz`
`=(-3)/(2)int(dz)/(z^(3))=(3)/(4z^(2))+C=(3)/(4)((2+x)/(2-x))^(2//3)+C`
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