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

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

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

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

- \frac{1}{e ^{x}} + ln ∣ x ∣ + C

Solution steps

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

Expand

\frac{x + e ^{x}}{x e ^{x}} : \frac{1}{e ^{x}} + \frac{1}{x}

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

= - \frac{1}{e ^{x}} + ln ∣ x ∣

Add a constant to the solution

= - \frac{1}{e ^{x}} + ln ∣ x ∣ + C

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