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

The integral of ((e^{-x})^2)/2 is -1/4 e^{-2x}+C

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

integral

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

- \frac{1}{4} e^{- 2 x} + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply u-substitution

= \frac{1}{2} \cdot \int - u d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} (- \int u d u)

Apply the Power Rule

= \frac{1}{2} (- \frac{u ^{2}}{2})

Substitute back

u = e^{- x}

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

Simplify

\frac{1}{2} (- \frac{( e ^{- x} ) ^{2}}{2}) : - \frac{1}{4} e^{- 2 x}

= - \frac{1}{4} e^{- 2 x}

Add a constant to the solution

= - \frac{1}{4} e^{- 2 x} + C

What is the integral of ((e^{-x})^2)/2 ?
The integral of ((e^{-x})^2)/2 is -1/4 e^{-2x}+C