What is the integral of (xe^{2x}+4e^x)/(xe^x) ?

The integral of (xe^{2x}+4e^x)/(xe^x) is e^x+4ln|x|+C

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integral of (xe^{2x}+4e^x)/(xe^x)

\int \frac{x e ^{2 x} + 4 e ^{x}}{x e ^{x}} d x

e^{x} + 4 ln ∣ x ∣ + C

Solution steps

\int \frac{x e ^{2 x} + 4 e ^{x}}{x e ^{x}} d x

Expand

\frac{x e ^{2 x} + 4 e ^{x}}{x e ^{x}} : e^{x} + \frac{4}{x}

= \int e^{x} + \frac{4}{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{4}{x} d x

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

\int \frac{4}{x} d x = 4 ln ∣ x ∣

= e^{x} + 4 ln ∣ x ∣

Add a constant to the solution

= e^{x} + 4 ln ∣ x ∣ + C

y = 4 x + 26, (- 1, 5)

v (t) = t^{2} e^{- 3 t}

\frac{d y}{d t} = \frac{t}{y 1 + t ^{2}}

\frac{5}{s ( s + 1 ) ( s + 2 )}

\frac{\partial}{\partial x} (- \frac{x}{8 y})

What is the integral of (xe^{2x}+4e^x)/(xe^x) ?
The integral of (xe^{2x}+4e^x)/(xe^x) is e^x+4ln|x|+C