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

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

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

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

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

Solution steps

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

Apply u-substitution

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

Expand

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

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

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

\int \frac{1}{u} d u = ln ∣ u ∣

= - \frac{2}{u} + ln ∣ u ∣

Substitute back

u = e^{x}

= - \frac{2}{e ^{x}} + ln ∣ e^{x} ∣

ln ∣ e^{x} ∣ = x

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

Add a constant to the solution

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

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