What is the integral of (3e^x+3)/(5e^{2x)} ?

The integral of (3e^x+3)/(5e^{2x)} is 1/5 (-3/(e^x)-3/2 e^{-2x})+C

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integral of (3e^x+3)/(5e^{2x)}

\int \frac{3 e ^{x} + 3}{5 e ^{2 x}} d x

\frac{1}{5} (- \frac{3}{e ^{x}} - \frac{3}{2} e^{- 2 x}) + C

Solution steps

\int \frac{3 e ^{x} + 3}{5 e ^{2 x}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{5} \cdot \int \frac{3 e ^{x} + 3}{e ^{2 x}} d x

Expand

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

= \frac{1}{5} \cdot \int \frac{3}{e ^{x}} + \frac{3}{e ^{2 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

= \frac{1}{5} (\int \frac{3}{e ^{x}} d x + \int \frac{3}{e ^{2 x}} d x)

\int \frac{3}{e ^{x}} d x = - \frac{3}{e ^{x}}

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

= \frac{1}{5} (- \frac{3}{e ^{x}} - \frac{3}{2} e^{- 2 x})

Add a constant to the solution

= \frac{1}{5} (- \frac{3}{e ^{x}} - \frac{3}{2} e^{- 2 x}) + C

What is the integral of (3e^x+3)/(5e^{2x)} ?
The integral of (3e^x+3)/(5e^{2x)} is 1/5 (-3/(e^x)-3/2 e^{-2x})+C