Question

What is the integral of `ln x / x^(1/2)`?

Answer 1

`int (lnx)/x^(1/2) dx = 2x^(1/2)(lnx -2)+C`

Explanation:

We can calculate the integral by parts:

`int (lnx)/x^(1/2) dx = 2int lnx d(x^(1/2)) = 2x^(1/2)lnx - 2int x^(1/2) d(lnx)`

Solving this last integral:

`int x^(1/2) d(lnx) = int x^(1/2) (dx)/x = int x^(-1/2)dx = 2x^(1/2)+C`

Putting it together:

`int (lnx)/x^(1/2) dx = 2x^(1/2)lnx -4x^(1/2)+C = 2x^(1/2)(lnx -2)+C`