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}) \cdot (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

y^{^{'}} + y \cdot tan (x) = \frac{1}{cos ( x )}

ln (1 + 12 x)

x + 16

\frac{\partial}{\partial x} (10 x yz)

\frac{d}{d x} (27 e^{3 x})

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