What is the integral of (e^{5x}+4)(e^{-2x}+15) ?

The integral of (e^{5x}+4)(e^{-2x}+15) is 1/3 e^{3x}+3e^{5x}-2e^{-2x}+60x+C

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integral of (e^{5x}+4)(e^{-2x}+15)

\int (e^{5 x} + 4) (e^{- 2 x} + 15) d x

\frac{1}{3} e^{3 x} + 3 e^{5 x} - 2 e^{- 2 x} + 60 x + C

Solution steps

\int (e^{5 x} + 4) (e^{- 2 x} + 15) d x

Expand

(e^{5 x} + 4) (e^{- 2 x} + 15) : e^{3 x} + 15 e^{5 x} + 4 e^{- 2 x} + 60

= \int e^{3 x} + 15 e^{5 x} + 4 e^{- 2 x} + 60 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^{3 x} d x + \int 15 e^{5 x} d x + \int 4 e^{- 2 x} d x + \int 60 d x

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

\int 15 e^{5 x} d x = 3 e^{5 x}

\int 4 e^{- 2 x} d x = - 2 e^{- 2 x}

\int 60 d x = 60 x

= \frac{1}{3} e^{3 x} + 3 e^{5 x} - 2 e^{- 2 x} + 60 x

Add a constant to the solution

= \frac{1}{3} e^{3 x} + 3 e^{5 x} - 2 e^{- 2 x} + 60 x + C

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What is the integral of (e^{5x}+4)(e^{-2x}+15) ?
The integral of (e^{5x}+4)(e^{-2x}+15) is 1/3 e^{3x}+3e^{5x}-2e^{-2x}+60x+C