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

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

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

\int \frac{( e ^{t} + 2 )}{e ^{t} + 2 t} d t

ln e^{t} + 2 t + C

Solution steps

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

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^{t} + 2 t

= ln e^{t} + 2 t

Add a constant to the solution

= ln e^{t} + 2 t + C

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