What is the integral of e^{(-x^2)/a} ?

The integral of e^{(-x^2)/a} is sqrt(a)(sqrt(pi))/2 \erf(x/(sqrt(a)))+C

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integral of e^{(-x^2)/a}

\int e^{\frac{- x ^{2}}{a}} d x

a \frac{π}{2} erf (\frac{x}{a}) + C

Solution steps

\int e^{\frac{- x ^{2}}{a}} d x

Simplify

= \int e^{- \frac{x ^{2}}{a}} d x

Apply exponent rule:

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

\frac{x ^{2}}{a} = (\frac{x}{a})^{2}

= \int e^{- (\frac{x}{a})^{2}} d x

Apply u-substitution

: \int e^{- u^{2}} a d u

= \int e^{- u^{2}} a d u

Take the constant out:

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

= a \cdot \int e^{- u^{2}} d u

Use the nonelementary integral:

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

= a \frac{π}{2} erf (u)

Substitute back

u = \frac{x}{a}

= a \frac{π}{2} erf (\frac{x}{a})

Add a constant to the solution

= a \frac{π}{2} erf (\frac{x}{a}) + C

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What is the integral of e^{(-x^2)/a} ?
The integral of e^{(-x^2)/a} is sqrt(a)(sqrt(pi))/2 \erf(x/(sqrt(a)))+C