Question

How do you find the integral `ln(x)/sqrtx`?

Answer 1

`int (ln(x))/sqrt(x)\ dx=2sqrt(x)ln(x)-4sqrt(x)+C`

Explanation:

Use integration-by-parts. Let `u=ln(x)` so that `du=1/x\ dx` and `dv=1/sqrt(x)\ dx=x^{-1/2}\ dx` so that `v=2x^{1/2}=2sqrt(x)`.

The integration-by-parts formula can be written in abstract form as: `int u\ dv=uv-int v\ du`.

For this problem, this becomes:

`int (ln(x))/sqrt(x)\ dx=2sqrt(x)ln(x)-int 2/sqrt(x)\ dx`

`=2sqrt(x)ln(x)-int 2x^{-1/2}\ dx`

`=2sqrt(x)ln(x)-4x^{1/2}+C`

`=2sqrt(x)ln(x)-4sqrt(x)+C`