What is the integral from 0 to 2 of pi(e^{-x^2})^2 ?

The integral from 0 to 2 of pi(e^{-x^2})^2 is 1.96857…

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integral from 0 to 2 of pi(e^{-x^2})^2

\int_{0}^{2} π (e^{- x^{2}})^{2} d x

1.96857 \dots

Solution steps

\int_{0}^{2} π (e^{- x^{2}})^{2} d x

Take the constant out:

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

= π \cdot \int_{0}^{2} (e^{- x^{2}})^{2} d x

Apply Integration By Parts

: [e^{- 2 x^{2}} x - \int - 4 e^{- 2 x^{2}} x^{2} d x]_{0}^{2}

= π [e^{- 2 x^{2}} x - \int - 4 e^{- 2 x^{2}} x^{2} d x]_{0}^{2}

\int - 4 e^{- 2 x^{2}} x^{2} d x = e^{- 2 x^{2}} x - \frac{π}{2 2} erf (2 x)

= π [e^{- 2 x^{2}} x - (e^{- 2 x^{2}} x - \frac{π}{2 2} erf (2 x))]_{0}^{2}

Simplify

= π [\frac{π}{2 2} erf (2 x)]_{0}^{2}

Compute the boundaries:

0.62661 \dots

= π 0.62661 \dots

Simplify

= 1.96857 \dots

f (x) = x^{- \frac{1}{4}}

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

e^{x}, at x = ln (7)

tan (e^{3 t}) + e^{t a n (3 t)}

f (x) = 7 e^{x} cos (x)

What is the integral from 0 to 2 of pi(e^{-x^2})^2 ?
The integral from 0 to 2 of pi(e^{-x^2})^2 is 1.96857…