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

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

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integral of (e^{-x}+e^{-3x})

\int (e^{- x} + e^{- 3 x}) d x

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

Solution steps

\int e^{- x} + e^{- 3 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 e^{- 3 x} d x

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

\int e^{- 3 x} d x = - \frac{1}{3} e^{- 3 x}

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

Add a constant to the solution

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

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