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

\frac{\partial}{\partial x} (2 x^{4} y + 4)

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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