What is the integral of (e^{-x}-e^3)(e^x-e^{-5}) ?

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

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integral of (e^{-x}-e^3)(e^x-e^{-5})

\int (e^{- x} - e^{3}) (e^{x} - e^{- 5}) d x

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

Solution steps

\int (e^{- x} - e^{3}) (e^{x} - e^{- 5}) d x

Expand

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

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

\int 1 d x = x

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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