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

The integral from 0 to x of e^{(-t^2)/2} is (pi/2)^{1/2}\erf(x/(sqrt(2)))

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

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

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

Solution steps

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

Simplify

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

Apply exponent rule:

\frac{a ^{n}}{b ^{n}} = (\frac{a}{b})^{n}

\frac{t ^{2}}{2} = (\frac{t}{2})^{2}

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

Apply u-substitution

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

Take the constant out:

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

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

Use the nonelementary integral:

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

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

Compute the boundaries:

\frac{π}{2} erf (\frac{x}{2})

= 2 \frac{π}{2} erf (\frac{x}{2})

Simplify

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

What is the integral from 0 to x of e^{(-t^2)/2} ?
The integral from 0 to x of e^{(-t^2)/2} is (pi/2)^{1/2}\erf(x/(sqrt(2)))