Compute int_0^2 1/(x-1)dx

vestirme4 2021-03-09 Answered
Compute 021x1dx
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Expert Answer

Adnaan Franks
Answered 2021-03-10 Author has 92 answers

021x1dx  is indeterminate since  y=1x1  has a vertical asymptote at  x=1, which lies between the upper and lower integration bounds.
To determine if the integral is convergent, you must then split the integral at x=1, and then use limits to evaluate:
021x1dx
=011x1dx+121x1dx (Rewrite as a sum)
limm10m1x1dx+limm1n21x1dx (Rewrite using limits)
limm1(ln|x1|)(0m)+limm1(ln|x1|)n2 (Integrate)
limm1(ln|m1|ln|01|)+limm1(ln|21|ln|n1|) (Evaluate)
limm1(ln|m1|ln1)+limm1(ln1ln|n1|) (Simplify)
limm1(ln|m1|)+limm1(ln|n1|) (Simplify using ln1=0)
limm1(ln|m1|)andlimm1(ln|n1|) aren't finite since ln0 is not defined.Therefore, the integral is divergent.

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