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

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

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integral of (e^x+2)/(e^x+2x)

\int \frac{e ^{x} + 2}{e ^{x} + 2 x} d x

ln ∣ e^{x} + 2 x ∣ + C

Solution steps

\int \frac{e ^{x} + 2}{e ^{x} + 2 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} + 2 x

= ln ∣ e^{x} + 2 x ∣

Add a constant to the solution

= ln ∣ e^{x} + 2 x ∣ + C

\int 2 + e^{x} d x

\int \frac{x ^{2} + 2 x + 2}{x ^{3} - 1} d x

\int \frac{1}{u ^{7}} d u

\int 3 x^{4} + x^{2} + 2 d x

\int 5 x^{2} + x d x

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