Evaluate the integrals \int_0^{\ln9}e^\theta(e^\theta-1)^{1/2}d\theta

arenceabigns 2021-10-22 Answered
Evaluate the integrals
0ln9eθ(eθ1)12dθ
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Expert Answer

wheezym
Answered 2021-10-23 Author has 103 answers
We have to evaluate the integral
0ln9eθ(eθ1)12dθ
let eθ1==t
on differentiating with respect to t we get
eθdθ=dt
dθ=dteθ
now the limits
t=eθ1
t=eln91
t=91
t=8
also
t=e01
t=11
t=0
now the definite integral
08eθt12dteθ
08t12dt
[t12+112+1]08
23[t32]08
23[8320]
23[23×32]
21123=32×2123
therefore
0ln9eθ(eθ1)12dθ=32×2123
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