What is the integral from 0 to 1 of e^{-2x^2} ?

The integral from 0 to 1 of e^{-2x^2} is 0.59814…

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integral from 0 to 1 of e^{-2x^2}

\int_{0}^{1} e^{- 2 x^{2}} d x

0.59814 \dots

Solution steps

\int_{0}^{1} e^{- 2 x^{2}} d x

= \int_{0}^{1} e^{- (2 x)^{2}} d x

Apply u-substitution

: \int_{0}^{2} e^{- u^{2}} \frac{1}{2} d u

= \int_{0}^{2} e^{- u^{2}} \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int_{0}^{2} e^{- u^{2}} d u

Use the nonelementary integral:

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

= \frac{1}{2} [\frac{π}{2} erf (u)]_{0}^{2}

Compute the boundaries:

0.84590 \dots

= \frac{1}{2} \cdot 0.84590 \dots

Simplify

= 0.59814 \dots

t \to 0 lim (- cos (t) sin (t))

x \to 1 lim (3 - x)

\int \frac{x ^{3}}{( 1 + x ^{4} ) ^{2}} d x

\frac{d y}{d x} (y) = (y)^{2} \cdot e^{3 x}

\int \frac{( 1 )}{( cos ^{(4)} ( x ) )} d x

What is the integral from 0 to 1 of e^{-2x^2} ?
The integral from 0 to 1 of e^{-2x^2} is 0.59814…