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

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

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

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

x - \frac{2}{e ^{x}} + C

Solution steps

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

Apply u-substitution

: \int \frac{u + 2}{u ^{2}} d u

= \int \frac{u + 2}{u ^{2}} d u

Expand

\frac{u + 2}{u ^{2}} : \frac{1}{u} + \frac{2}{u ^{2}}

= \int \frac{1}{u} + \frac{2}{u ^{2}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{u} d u + \int \frac{2}{u ^{2}} d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{2}{u ^{2}} d u = - \frac{2}{u}

= ln ∣ u ∣ - \frac{2}{u}

Substitute back

u = e^{x}

= ln ∣ e^{x} ∣ - \frac{2}{e ^{x}}

ln ∣ e^{x} ∣ = x

= x - \frac{2}{e ^{x}}

Add a constant to the solution

= x - \frac{2}{e ^{x}} + C

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