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The integral int(2x^(3)-1)/(x^(4)+x)dx ...

The integral `int(2x^(3)-1)/(x^(4)+x)dx` is equal to :
(Here `C` is a constant of integration)

A

`(1)/(2)log_(e)((x^(3)+1)^(2))/(|x^(3)|)+c`

B

`log_(e)(|x^(3)+1|)/(x^(2))+c`

C

`(1)/(2)log_(e)(|x^(3)+1|)/(x^(2))+C`

D

`log_(e)|(x^(3)+1)/(x)|+C`

Text Solution

Verified by Experts

The correct Answer is:
D
`int(2x^(3)-1)/(x^(4)+x)=int(2x^(3))/(x^(4)+x)-int(1)/(x^(4)+x)dx=(2)/(3)int(3x^(2))/(x^(3)+x)-int(1)/(x.(x^(3)+1))xx(x^(2))/(x^(2))dx=(2)/(3)log(x^(3)H)-(1)/(3)int(dt)/(t(t+1))`
`("put "x^(3)=t)`
`(2)/(3)log(x(3)+1)-(1)/(3)int((1)/(t)-(1)/(t+1))dt`
`(2)/(3)log(x^(3)+1)-(1)/(3)[logt-log(t+1)]+c=(2)/(3)log(x^(3)+1)-(1)/(3)log((x^(3))/(1+x^(3)))+c=(1)/(3)log|(x^(3)+1)^(2)(x^(3))/(1+x^(3))|+c`
`(1)/(3)log|((1+x^(3))/(x^(3)))^(3)|+c," "log|(1+x^(3))/(x)|`
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