int(x^3dx)/(sqrt(1+x^2))i se q u a lto ...

`int(x^3dx)/(sqrt(1+x^2))i se q u a lto` `1/3sqrt(1+x^2)(2+x^2)+C` `1/3sqrt(1+x^2)(x^2-1)+C` `1/3(1+x^2)^(3/2)+C` (d) `1/3sqrt(1+x^2)(x^2-2)+C`

A

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

B

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

C

`(1)/(3)(1+x^(2))^(3//2)+C`

D

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

Text Solution

Verified by Experts

The correct Answer is:

D

`I=int (x^(3)dx)/(sqrt(1+x^(2)))=int (x xxx^(2)dx)/(sqrt(1+x^(2)))`
Let `t=sqrt(1+x^(2))`
or `(dt)/(dx)=(x)/(sqrt(1+x^(2)))`
` :. I=int(t^(2)-1)dt`
`=(t^(3))/(3)-t+C=(t)/(3)(t^(2)-3)+C`
`=(1)/(3)sqrt(1+x^(2))(x^(2)-2)+C`

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