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

The integral of (2+e^{-x})/(x+2) is 2ln|x+2|+e^2\Ei(-(x+2))+C

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integral of (2+e^{-x})/(x+2)

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

2 ln ∣ x + 2 ∣ + e^{2} Ei (- (x + 2)) + C

Solution steps

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

Apply u-substitution

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

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

Expand

\frac{2 + e ^{- u + 2}}{u} : \frac{2}{u} + \frac{e ^{- u + 2}}{u}

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

\int \frac{2}{u} d u = 2 ln ∣ u ∣

\int \frac{e ^{- u + 2}}{u} d u = e^{2} Ei (- u)

= 2 ln ∣ u ∣ + e^{2} Ei (- u)

Substitute back

u = x + 2

= 2 ln ∣ x + 2 ∣ + e^{2} Ei (- (x + 2))

Add a constant to the solution

= 2 ln ∣ x + 2 ∣ + e^{2} Ei (- (x + 2)) + C

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What is the integral of (2+e^{-x})/(x+2) ?
The integral of (2+e^{-x})/(x+2) is 2ln|x+2|+e^2\Ei(-(x+2))+C