Evaluate the following definite integral. \int_{1}^{e}\ln 2 xdx

ruigE 2020-12-28 Answered
Evaluate the following definite integral.
1eln2xdx
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Expert Answer

Clelioo
Answered 2020-12-29 Author has 88 answers
1eln2xdx
Substitute u =2x
dudx=2
dx=12du
=1222eln(u)du
Now solve 22elnudu
Integrate by parts:
fg=fgfg
f=lnu,f=1u
g'=1, g=u
[uln(u)]22e22e1du
Now solve 22e1du=[u]22e
Plug in the solved integrals:
[uln(u)]22e22e1du
=[uln(u)u]22e
Plug in the solved integrals:
1222eln(u)du
=[uln(u)2u2]22e
Undo the substitution u=2x
=[xln(2x)x]1e
=[eln(2e)e][1ln(2)1]
=eln(2e)eln(2)+1
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Jeffrey Jordon
Answered 2021-11-05 Author has 2047 answers

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