Question

`int sqrt(x^2+5/x)=`?

Answer 1

`sqrt(x^2+5) +5/(2sqrt5) ln((sqrt(x^2+5)-sqrt5)/(sqrt(x^2+5)+sqrt5))+C`

Explanation:

Substitute `x^2+5=u^2`. Then, `2xdx=2udu` and thus `dx/x =(udu)/x^2=(udu)/(u^2-5)`.

So, our integral becomes

`int sqrt(x^2+5)/x dx = int u (udu)/(u^2-5)`
`qquad = int (u^2-5+5)/(u^2-5) = int (1+5/(u^2-5))du`
`qquad = u+5 int (du)/(u^2-(sqrt5)^2)= u + 5/(2sqrt5) ln((u-sqrt5)/(u+sqrt5))`
`qquad = sqrt(x^2+5) +5/(2sqrt5) ln((sqrt(x^2+5)-sqrt5)/(sqrt(x^2+5)+sqrt5))`