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

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

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

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

x + e^{- 2 x} + C

Solution steps

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

Expand

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

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

\int 1 d x = x

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

= x - (- e^{- 2 x})

Simplify

= x + e^{- 2 x}

Add a constant to the solution

= x + e^{- 2 x} + C

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