Get the answer to "Evaluate the following integrals. ∫1e1dxx(ln2x+2ln⁡x+2)"

SchachtN

Answered question

2021-10-14

Evaluate the following integrals.

\int_{\frac{1}{e}}^{1} \frac{d x}{x (\ln^{2} x + 2 \ln x + 2)}

Pohanginah

Skilled2021-10-15Added 96 answers

We have

I = \int_{\frac{1}{e}}^{1} \frac{d x}{x (\ln^{2} x + 2 \ln x + 2)}

Let

\ln x = u

and limits

u \to - 1

to 0

\frac{1}{x} d x = d u

put in equation (1)

I = \int_{- 1}^{0} \frac{d u}{(u^{2} + 2 u + 2)}

= \int_{- 1}^{0} \frac{d u}{| {(u + 1)}^{2} + 1 |}

Let

u + 1 = t

and limits

t \to 0

to 1
put in equation

I = \int_{0}^{1} \frac{d t}{t^{2} + 1}

= {[\tan^{- 1} t]}_{0}^{1}

= [\tan^{- 1} 1 - \tan^{- 1} 0]

= \frac{π}{4}

Hence,

I = \frac{π}{4}

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