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

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

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integral of e^{2x}+1/(e^{5x)}

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

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

Solution steps

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

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

\int \frac{1}{e ^{5 x}} d x = - \frac{1}{5} e^{- 5 x}

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

Add a constant to the solution

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

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