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

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

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

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

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

Solution steps

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

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

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

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

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

Simplify

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

Add a constant to the solution

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

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