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

The integral from 0 to x^2 of e^{-2t^2} is (sqrt(pi))/(2sqrt(2))\erf(sqrt(2)x^2)

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

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

\frac{π}{2 2} erf (2 x^{2})

Solution steps

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

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

Apply u-substitution

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

= \int_{0}^{2 x^{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 x^{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 x^{2}}

Compute the boundaries:

\frac{π}{2} erf (2 x^{2})

= \frac{1}{2} \cdot \frac{π}{2} erf (2 x^{2})

Simplify

= \frac{π}{2 2} erf (2 x^{2})

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What is the integral from 0 to x^2 of e^{-2t^2} ?
The integral from 0 to x^2 of e^{-2t^2} is (sqrt(pi))/(2sqrt(2))\erf(sqrt(2)x^2)