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

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

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integral of (xe^x+5)/x

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

e^{x} + 5 ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

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

\int \frac{5}{x} d x = 5 ln ∣ x ∣

= e^{x} + 5 ln ∣ x ∣

Add a constant to the solution

= e^{x} + 5 ln ∣ x ∣ + C

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