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

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

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

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

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

Solution steps

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

Expand

(e^{- x} - e^{2}) (e^{x} + e^{- 5}) : 1 + e^{- x - 5} - e^{2 + x} - \frac{1}{e ^{3}}

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

\int 1 d x = x

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

\int e^{2 + x} d x = e^{2 + x}

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

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

Simplify

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

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

Add a constant to the solution

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

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