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

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

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integral of ((e^x+1))/(e^x+x)

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

ln ∣ e^{x} + x ∣ + C

Solution steps

\int \frac{e ^{x} + 1}{e ^{x} + x} d x

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = e^{x} + x

= ln ∣ e^{x} + x ∣

Add a constant to the solution

= ln ∣ e^{x} + x ∣ + C

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