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

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

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integral of e^{3x}(3^x-e^{-2x})

\int e^{3 x} (3^{x} - e^{- 2 x}) d x

\frac{1}{3 + ln ( 3 )} \cdot 3^{\frac{( 3 + l n ( 3 ) ) x}{l n ( 3 )}} - e^{x} + C

Solution steps

\int e^{3 x} (3^{x} - e^{- 2 x}) d x

Expand

e^{3 x} (3^{x} - e^{- 2 x}) : 3^{x} e^{3 x} - e^{x}

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

\int 3^{x} e^{3 x} d x = \frac{1}{3 + ln ( 3 )} \cdot 3^{\frac{( 3 + l n ( 3 ) ) x}{l n ( 3 )}}

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

= \frac{1}{3 + ln ( 3 )} \cdot 3^{\frac{( 3 + l n ( 3 ) ) x}{l n ( 3 )}} - e^{x}

Add a constant to the solution

= \frac{1}{3 + ln ( 3 )} \cdot 3^{\frac{( 3 + l n ( 3 ) ) x}{l n ( 3 )}} - e^{x} + C

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