First let's solve the indefinite integral intln^2(x)/sqrt(x)dx∫ln2(x)√xdx. by applying integration by parts twice:
First Integration by Parts
Let u = ln^2(x)u=ln2(x) and dv = 1/sqrt(x)dxdv=1√xdx
Then du = (2ln(x))/xdu=2ln(x)x and v = 2sqrt(x)v=2√x
Thus
intln^2(x)/sqrt(x)dx = intudv∫ln2(x)√xdx=∫udv
= uv - intvdu=uv−∫vdu
=2sqrt(x)ln^2(x) - 4intln(x)/sqrt(x)dx=2√xln2(x)−4∫ln(x)√xdx
Second Integration by Parts
Let u = ln(x)u=ln(x) and dv = 1/sqrt(x)dv=1√x
Then du = 1/xdu=1x and v = 2sqrt(x)v=2√x
Thus
intln(x)/sqrt(x)dx = intudv∫ln(x)√xdx=∫udv
= uv - intvdu=uv−∫vdu
= 2sqrt(x)ln(x) - 2int1/sqrt(x)dx=2√xln(x)−2∫1√xdx
= 2sqrt(x)ln(x) - 4sqrt(x) + C=2√xln(x)−4√x+C
Putting it together, we get
intln^2(x)/sqrt(x)dx = 2sqrt(x)ln^2(x) - 4intln(x)/sqrt(x)dx∫ln2(x)√xdx=2√xln2(x)−4∫ln(x)√xdx
= 2sqrt(x)ln^2(x) - 4(2sqrt(x)ln(x) - 4sqrt(x) + C)=2√xln2(x)−4(2√xln(x)−4√x+C)
= 2sqrt(x)(ln^2(x) - 4ln(x) + 8) + C=2√x(ln2(x)−4ln(x)+8)+C
Now we can evaluate the original definite integral.
int_0^piln^2(x)/sqrt(x)dx = [2sqrt(x)(ln^2(x) - 4ln(x) + 8)]_0^pi∫π0ln2(x)√xdx=[2√x(ln2(x)−4ln(x)+8)]π0
(Note that as there is a discontinuity at 00 we must evaluate this using a limit)
= 2sqrt(pi)(ln^2(pi) - 4ln(pi) + 8) - lim_(x->0)2sqrt(x)(ln^2(x) - 4ln(x) + 8)
= 2sqrt(pi)(ln^2(pi) - 4ln(pi) + 8) - 0
= 2sqrt(pi)(ln^2(pi) - 4ln(pi) + 8)
