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…

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

= 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

d er i v a t i v e f (x) = 7 e^{x} cos (x)

\int 6 cot^{4} (x) d x

y^{^{''}} - 4 y = e^{- 2 x} - 2 x, y (0) = 0, y^{^{'}} (0) = 0

y^{^{''}} + y = \frac{3}{2} sin (x)

\int x + 1 x d 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…