What is the integral of (x-1)/(ln(x)) ?

The integral of (x-1)/(ln(x)) is \Ei(2ln(x))-\Ei(ln(x))+C

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integral of (x-1)/(ln(x))

\int \frac{x - 1}{ln ( x )} d x

Ei (2 ln (x)) - Ei (ln (x)) + C

Solution steps

\int \frac{x - 1}{ln ( x )} d x

Expand

\frac{x - 1}{ln ( x )} : \frac{x}{ln ( x )} - \frac{1}{ln ( x )}

= \int \frac{x}{ln ( x )} - \frac{1}{ln ( x )} d x

Apply the Sum Rule:

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

= \int \frac{x}{ln ( x )} d x - \int \frac{1}{ln ( x )} d x

\int \frac{x}{ln ( x )} d x = Ei (2 ln (x))

\int \frac{1}{ln ( x )} d x = Ei (ln (x))

= Ei (2 ln (x)) - Ei (ln (x))

Add a constant to the solution

= Ei (2 ln (x)) - Ei (ln (x)) + C

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What is the integral of (x-1)/(ln(x)) ?
The integral of (x-1)/(ln(x)) is \Ei(2ln(x))-\Ei(ln(x))+C