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The value of int(dx)/(xsqrt(1-x^(3))) is...

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

A

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

B

`(1)/(2)log|(sqrt(1-x^(2))+1)/(sqrt1-x^(2))-1|+C`

C

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

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
a
We have , `I=int(1)/(xsqrt(1-x^(3)))dx`
`rArrI=-(1)/(3)int(1)/(x^(3)sqrt(1-x^(3)))(-3x^(2))dx`
`rArrI=-(1)/(3)int(1)/(x^(3)sqrt(1-x^(3)))(1-x^(3))dx`
`rArrI=-(1)/(3)int(1)/((1-t^(2))sqrt(t^(2)))` 2t dt , where `t^(2)=1-x^(3)`
`rArrI=-(2)/(3)int(1)/(1-t^(2))dt=(2)/(3)int(1)/(t^(2)-1^(2))dt=(1)/(3)log|(t-1)/(t+1)|+C`
`rArrI=(1)/(3)log|(sqrt(1-x^(3))-1)/(sqrt(1-x^(3))+1)|+C`
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