What is the improper integrals sqrtx ln 5x dx from 1 to e ?

1 Answer
May 11, 2018

=2/3e^(3/2)ln(5e)-4/9e^(3/2)-2/3ln(5)+4/9

Explanation:

In preparation for evaluating the definite integral, we should first find the antiderivative intsqrtxln(5x)dx, which can be solved using Integration by Parts:

u=ln(5x)
du=x^-1dx
dv=sqrtxdx
v=2/3x^(3/2)

uv-intvdu=2/3x^(3/2)ln(5x)-2/3intx^(3/2)x^(-1)dx

#=2/3x^(3/2)ln(5x)-2/3intsqrtxdx#

=2/3x^(3/2)ln(5x)-4/9x^(3/2) (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

This is not an improper integral; the integrand sqrt(x)ln(5x) is continuous on the interval of integration [1, e].

Thus,

int_1^esqrtxln(5x)dx=[2/3x^(3/2)ln(5x)-4/9x^(3/2)]|_1^e

=2/3e^(3/2)ln(5e)-4/9e^(3/2)-2/3ln(5)+4/9