Question

Integrate :-
$\displaystyle \bf \large\int\limits^1_0 \dfrac{log (x + 1)}{x} dx$
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Answer 1

Answer:

$\displaystyle \sf \int\limits^1_0 {\dfrac{log(x+1)}{x}} \, dx$

To proceed further with this, substitute u=-x.

$-{\displaystyle\int} \sf -\dfrac{\ln\left(1-u\right)}{u}\, du$ ----(1)

This is the Spence's function!

It is a special integral!

${Li _{2}(z)=\displaystyle \sf -\int _{0}^{z}{\ln(1-u) \over u}\,du{\text{, }}z\in \mathbb {C} }$

So, we can rewrite equation (1). We get:

$\sf -Li_2\left(u\right)$

After substituting u = -x, we get:

$\sf -Li_2\left(-x\right)+constant$

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Answer 2

$\huge\mathcal{\fcolorbox{cyan}{black}{\pink{ANSWER}}}$

See the attachment please.

It is a special type of integration.