What is the integral of e^x(1-(e^{-x})/x) ?

The integral of e^x(1-(e^{-x})/x) is e^x-ln|x|+C

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integral of e^x(1-(e^{-x})/x)

\int e^{x} (1 - \frac{e ^{- x}}{x}) d x

e^{x} - ln ∣ x ∣ + C

Solution steps

\int e^{x} (1 - \frac{e ^{- x}}{x}) d x

Expand

e^{x} (1 - \frac{e ^{- x}}{x}) : e^{x} - \frac{1}{x}

= \int e^{x} - \frac{1}{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 e^{x} d x - \int \frac{1}{x} d x

\int e^{x} d x = e^{x}

\int \frac{1}{x} d x = ln ∣ x ∣

= e^{x} - ln ∣ x ∣

Add a constant to the solution

= e^{x} - ln ∣ x ∣ + C

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What is the integral of e^x(1-(e^{-x})/x) ?
The integral of e^x(1-(e^{-x})/x) is e^x-ln|x|+C