If `int1/(xsqrt(1-x^3))dx=alog|(sqrt(1-x^3)-1)/(sqrt(1-x^3)+1)|+b ,t h e nai se q u a l` `1/3` (b) `2/3` (c) `-1/3` (d0 `-2/3`

Options

A. `1/3`
B. `2/3`
C. `-1/3`
D. `-2/3`

Answer

B

Explanation

Let `l=int(x^(2))/(x^(3)sqrt(1-x^(3)))dx`
Put `1-x^(3)=t^(2)implies-3x^(2)dx=2tdt`
`therefore -2/3int(tdt)/((1-t^(2))t)=-(2)/(3)int (dt)/(1-t^(2))`
`=-2/3log ((1+t)/(1-t))+b`
`=2/3log |(1-sqrt(1-x^(3)))/(1+sqrt(1-x^(2)))|+b`
`therefore a=2/3`