Evaluate the following integrals. \int_{1/e}^1\frac{dx}{x(\ln^2x+2\ln x+2

SchachtN 2021-10-14 Answered
Evaluate the following integrals.
1e1dxx(ln2x+2lnx+2)
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Expert Answer

Pohanginah
Answered 2021-10-15 Author has 96 answers
We have
I=1e1dxx(ln2x+2lnx+2)
Let
lnx=u and limits u1 to 0
1xdx=du
put in equation (1)
I=10du(u2+2u+2)
=10du|(u+1)2+1|
Let
u+1=t and limits t0 to 1
put in equation
I=01dtt2+1
=[tan1t]01
=[tan11tan10]
=π4
Hence, I=π4
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