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

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

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

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

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

Solution steps

\int \frac{e ^{x} - e ^{- x}}{e ^{2 x}} d x

Expand

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

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

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

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

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

Simplify

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

Add a constant to the solution

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

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x \to \infty lim (e^{5 x l n (1 + \frac{1}{x})})

\int \frac{4 x ^{3} - 5 x}{x} d x

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