In preparation for evaluating the definite integral, we should first find the antiderivative intsqrtxln(5x)dx∫√xln(5x)dx, which can be solved using Integration by Parts:
u=ln(5x)u=ln(5x)
du=x^-1dxdu=x−1dx
dv=sqrtxdxdv=√xdx
v=2/3x^(3/2)v=23x32
uv-intvdu=2/3x^(3/2)ln(5x)-2/3intx^(3/2)x^(-1)dxuv−∫vdu=23x32ln(5x)−23∫x32x−1dx
#=2/3x^(3/2)ln(5x)-2/3intsqrtxdx#
=2/3x^(3/2)ln(5x)-4/9x^(3/2)=23x32ln(5x)−49x32 (leaving out the constant as we're going to use this to evaluate a definite integral)
Now, we may evaluate the improper definite integral:
int_1^esqrtxln(5x)dx∫e1√xln(5x)dx
This is not an improper integral; the integrand sqrt(x)ln(5x)√xln(5x) is continuous on the interval of integration [1, e][1,e].
Thus,
int_1^esqrtxln(5x)dx=[2/3x^(3/2)ln(5x)-4/9x^(3/2)]|_1^e∫e1√xln(5x)dx=[23x32ln(5x)−49x32]∣∣∣e1
=2/3e^(3/2)ln(5e)-4/9e^(3/2)-2/3ln(5)+4/9=23e32ln(5e)−49e32−23ln(5)+49