Question

What' the integral of ln √x dx ? (With the substitution)

Answer 1

`intlnsqrtxdx=(xlnx)/2-x/2+C`

Explanation:

So, we want to determine

`intlnsqrtxdx`. We can rewrite as

`intln(x^(1/2))dx=1/2intlnxdx`, as `ln(x^a)=alnx`

Now, we can use Integration by Parts, making the following selections:

`u=lnx`

`du=x^-1dx`

`dv=dx`

`v=intdx=x`

`uv-intvdu=xlnx-intx^-1(x)dx=xlnx-intdx=xlnx-x+C`

Recalling that we must multiply by a factor of `1/2`:

`intlnsqrtxdx=(xlnx)/2-x/2+C`