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

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

A

`(1)/(2)"log"_(e)(|x^(3)+1|)/(x^(2))+C`

B

`(1)/(2)"log"_(e)(|x^(3)+1|^(2))/(|x^(3)|)+C`

C

`"log"_(e)|(x^(3)+1)/(x)|+C`

D

`"log"_(e)(|x^(3)+1|)/(x^(2))+C`

Text Solution

Verified by Experts

The correct Answer is:
C
Key Idea
(i) Divide each term of numerator and denominator by `x^(2)`.
(ii) Let `x^(2)+(1)/(x) = t`
Let integral is `I = int(2x^(3)-1)/(x^(4)+x)dx = int(2x-1//x^(2))/(x^(2)+(1)/(x))dx` [dividing each term of numerator and denominator by `x^(2)`]
Put `x^(2)+(1)/(x)=t rArr (2x+(-(1)/(x^(2))))dx = dt`
`therefore I = int(dt)/(t)=log_(e)|(t)|+C`
`=log_(e)|(x^(2)+(1)/(x))|+C`
`=log_(e)|(x^(3)+1)/(x)|+C`
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