What is the integral from 2 to 3 of x/(ln(x-1)) ?

The integral from 2 to 3 of x/(ln(x-1)) is -2\Ei(0)+\Ei(ln(2))+\Ei(2ln(2))

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integral from 2 to 3 of x/(ln(x-1))

\int_{2}^{3} \frac{x}{ln ( x - 1 )} d x

- 2 Ei (0) + Ei (ln (2)) + Ei (2 ln (2))

Solution steps

\int_{2}^{3} \frac{x}{ln ( x - 1 )} d x

Apply u-substitution

: \int_{1}^{2} \frac{u + 1}{ln ( u )} d u

= \int_{1}^{2} \frac{u + 1}{ln ( u )} d u

Expand

\frac{u + 1}{ln ( u )} : \frac{u}{ln ( u )} + \frac{1}{ln ( u )}

= \int_{1}^{2} \frac{u}{ln ( u )} + \frac{1}{ln ( u )} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{1}^{2} \frac{u}{ln ( u )} d u + \int_{1}^{2} \frac{1}{ln ( u )} d u

\int_{1}^{2} \frac{u}{ln ( u )} d u = Ei (2 ln (2)) - Ei (0)

\int_{1}^{2} \frac{1}{ln ( u )} d u = Ei (ln (2)) - Ei (0)

= Ei (2 ln (2)) - Ei (0) + Ei (ln (2)) - Ei (0)

Simplify

= - 2 Ei (0) + Ei (ln (2)) + Ei (2 ln (2))

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What is the integral from 2 to 3 of x/(ln(x-1)) ?
The integral from 2 to 3 of x/(ln(x-1)) is -2\Ei(0)+\Ei(ln(2))+\Ei(2ln(2))