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

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

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

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

2 ln ∣ x ∣ + Ei (- x) + C

Solution steps

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

Expand

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

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{e ^{- x}}{x} d x = Ei (- x)

= 2 ln ∣ x ∣ + Ei (- x)

Add a constant to the solution

= 2 ln ∣ x ∣ + Ei (- x) + C

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