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int(1)/(xsqrt(1-x^(3)))dx is equal to...

`int(1)/(xsqrt(1-x^(3)))`dx is equal to

A

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

B

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

C

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

D

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

Text Solution

Verified by Experts

The correct Answer is:
D
`l=(dx)/(xsqrt(1-x)^(3))`
On multiplying numerator and denominator by `x^(2)`, we get
`l=int(x^(2)dx)/(x^(3)sqrt(1-x^(3)))`
On putting `1-x^(3)=t^(2)=t = sqrt(1-x^(3))`
`rArr" "-3x^(2)dx=2t dt`
`rArr" "x^(2)dx=-(2)/(3)t dt and x^(3)=1-t^(2)`
`therefore" "l=-(2)/(3)int(tdt)/((1-t^(2))t)`
`=(2)/(3)int(dt)/(t^(2)-1)=(2)/(3)[(1)/(2)log|(t-1)/(t+1)|]+C`
`=(1)/(3)log|(t-1)/(t+1)|+C`
`=(1)/(3)log|(sqrt(1-x^(3))-1)/(sqrt(1-x^(3))+1|+C`
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