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

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

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

\int π (e^{- x})^{2} d x

- \frac{π}{2} e^{- 2 x} + C

Solution steps

\int π (e^{- x})^{2} d x

Take the constant out:

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

= π \cdot \int (e^{- x})^{2} d x

Apply u-substitution

: \int - u d u

= π \cdot \int - u d u

Take the constant out:

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

= π (- \int u d u)

Apply the Power Rule

: \frac{u ^{2}}{2}

= π (- \frac{u ^{2}}{2})

Substitute back

u = e^{- x}

= π (- \frac{( e ^{- x} ) ^{2}}{2})

Simplify

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

= - \frac{π}{2} e^{- 2 x}

Add a constant to the solution

= - \frac{π}{2} e^{- 2 x} + C

x cos (\frac{y}{x}) \frac{d y}{d x} = y cos (\frac{y}{x}) - x

\frac{\partial}{\partial x} (\frac{2}{x ^{2}} + y)

- 4 x^{2} + x - 25, - 3 x^{2} + 17 x + 38

\frac{95 t}{99 t - 95}

\int \frac{2 x}{x ^{2} + 100} d x

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