What is the integral of (5-xe^x)/x ?

The integral of (5-xe^x)/x is 5ln|x|-e^x+C

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integral of (5-xe^x)/x

\int \frac{5 - x e ^{x}}{x} d x

5 ln ∣ x ∣ - e^{x} + C

Solution steps

\int \frac{5 - x e ^{x}}{x} d x

Expand

\frac{5 - x e ^{x}}{x} : \frac{5}{x} - e^{x}

= \int \frac{5}{x} - e^{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{5}{x} d x - \int e^{x} d x

\int \frac{5}{x} d x = 5 ln ∣ x ∣

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

= 5 ln ∣ x ∣ - e^{x}

Add a constant to the solution

= 5 ln ∣ x ∣ - e^{x} + C

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