What is the integral of (e^{3x}+3e^x+2)/(e^x) ?

The integral of (e^{3x}+3e^x+2)/(e^x) is 1/2 e^{2x}+3x-2/(e^x)+C

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integral of (e^{3x}+3e^x+2)/(e^x)

\int \frac{e ^{3 x} + 3 e ^{x} + 2}{e ^{x}} d x

\frac{1}{2} e^{2 x} + 3 x - \frac{2}{e ^{x}} + C

Solution steps

\int \frac{e ^{3 x} + 3 e ^{x} + 2}{e ^{x}} d x

Expand

\frac{e ^{3 x} + 3 e ^{x} + 2}{e ^{x}} : e^{2 x} + 3 + \frac{2}{e ^{x}}

= \int e^{2 x} + 3 + \frac{2}{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 e^{2 x} d x + \int 3 d x + \int \frac{2}{e ^{x}} d x

\int e^{2 x} d x = \frac{1}{2} e^{2 x}

\int 3 d x = 3 x

\int \frac{2}{e ^{x}} d x = - \frac{2}{e ^{x}}

= \frac{1}{2} e^{2 x} + 3 x - \frac{2}{e ^{x}}

Add a constant to the solution

= \frac{1}{2} e^{2 x} + 3 x - \frac{2}{e ^{x}} + C

f (t) = - \frac{1}{t ^{2}}

\int 2 arctan (3 x) d x

\int - x e^{- l n (x)} d x

f (x) = 2 x^{3} + 3 x^{2}, at x = 1

\int x^{2} f (x) d x

What is the integral of (e^{3x}+3e^x+2)/(e^x) ?
The integral of (e^{3x}+3e^x+2)/(e^x) is 1/2 e^{2x}+3x-2/(e^x)+C