What is the integral of (2(e^x-1))/(e^x-x) ?

The integral of (2(e^x-1))/(e^x-x) is 2ln|e^x-x|+C

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integral of (2(e^x-1))/(e^x-x)

\int \frac{2 ( e ^{x} - 1 )}{e ^{x} - x} d x

2 ln ∣ e^{x} - x ∣ + C

Solution steps

\int \frac{2 ( e ^{x} - 1 )}{e ^{x} - x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{e ^{x} - 1}{e ^{x} - x} d x

Apply u-substitution

: \int \frac{1}{u} d u

= 2 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 ln ∣ u ∣

Substitute back

u = e^{x} - x

= 2 ln ∣ e^{x} - x ∣

Add a constant to the solution

= 2 ln ∣ e^{x} - x ∣ + C

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What is the integral of (2(e^x-1))/(e^x-x) ?
The integral of (2(e^x-1))/(e^x-x) is 2ln|e^x-x|+C