How do you evaluate the definite integral #int dx / ( x(sqrt(ln(x)))# from #[e, e^81]#?
2 Answers
Explanation:
We want to find:
#int_e^(e^81)dx/(xsqrtln(x))#
We will want to use substitution. Note that if we let
Before making these substitutions, note that making the substitutions will necessitate that the boundaries change. To change them, plug the current bounds into
We see that
Thus, since
#int_e^(e^81)color(blue)dx/(color(blue)xsqrtcolor(red)ln(x))=int_1^81(du)/sqrtu#
Rewrite the integral:
#int_1^81(du)/sqrtu=int_1^81u^(-1/2)du#
Which can be integrated using the rule:
#int_a^bu^ndu=[u^(n+1)/(n+1)]_a^b" "" ",n!=-1#
So:
#int_1^81u^(-1/2)du=[u^(-1/2+1)/(-1/2+1)]_1^81=[2sqrtu]_1^81=2sqrt81-2sqrt1=16#
16
Explanation:
just as an alternative approach,
for
if you recognise the pattern
then the integral is