First let's solve the indefinite integral intln^2(x)/sqrt(x)dx. by applying integration by parts twice:
First Integration by Parts
Let u = ln^2(x) and dv = 1/sqrt(x)dx
Then du = (2ln(x))/x and v = 2sqrt(x)
Thus
intln^2(x)/sqrt(x)dx = intudv
= uv - intvdu
=2sqrt(x)ln^2(x) - 4intln(x)/sqrt(x)dx
Second Integration by Parts
Let u = ln(x) and dv = 1/sqrt(x)
Then du = 1/x and v = 2sqrt(x)
Thus
intln(x)/sqrt(x)dx = intudv
= uv - intvdu
= 2sqrt(x)ln(x) - 2int1/sqrt(x)dx
= 2sqrt(x)ln(x) - 4sqrt(x) + C
Putting it together, we get
intln^2(x)/sqrt(x)dx = 2sqrt(x)ln^2(x) - 4intln(x)/sqrt(x)dx
= 2sqrt(x)ln^2(x) - 4(2sqrt(x)ln(x) - 4sqrt(x) + C)
= 2sqrt(x)(ln^2(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
(Note that as there is a discontinuity at 0 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)
