Question

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

Answer 1

`=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`