Evaluate : `int (2xdx)/((x^(2)+1)^(3//2)`

Explanation

Method 1
`"Let" " "I=int(2xdx)/(3sqrt(x^(2)+1))`
`"Let"u=x^(2)+1, "then" " " du=2xdx`.
`I = int u^(-1//3) du` [In the form of ` u^(n)` du]
`=(u^(2//3))/(2//3)+C` [Integrate with respect to u]
`=(3)/(2) u^(2//3)+C`
`(x^(2)+1)^(2//3)+C` [Replace u by ` x^(2)+1]`
Method 2
Let ` u=3sqrt(x^(2)+1) rArr u^(3)=x^(2)+1`
Then `3u^(2)du=2xdx`
`I = int (2xdx)/(3 sqrt(x^(2)+1))`
`=int(3u^(2)du)/(u)`
`=3.int u du`
` =3.(u^(2))/(2)+C` [ Intergrate with respect to u ]
`=(3)/(2) (x^(2)+1)^(2//3)+C` [Replace u by ` (x^(2)+1)^(1//3)`]