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

1 Answer
VNVDVI
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#

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