What is the integral of e^{(-x^2)/5} ?

The integral of e^{(-x^2)/5} is sqrt(5)(sqrt(pi))/2 \erf(x/(sqrt(5)))+C

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integral of e^{(-x^2)/5}

\int e^{\frac{- x ^{2}}{5}} d x

5 \frac{π}{2} erf (\frac{x}{5}) + C

Solution steps

\int e^{\frac{- x ^{2}}{5}} d x

Simplify

= \int e^{- \frac{x ^{2}}{5}} d x

Apply exponent rule:

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

\frac{x ^{2}}{5} = (\frac{x}{5})^{2}

= \int e^{- (\frac{x}{5})^{2}} d x

Apply u-substitution

: \int e^{- u^{2}} 5 d u

= \int e^{- u^{2}} 5 d u

Take the constant out:

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

= 5 \cdot \int e^{- u^{2}} d u

Use the nonelementary integral:

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

= 5 \frac{π}{2} erf (u)

Substitute back

u = \frac{x}{5}

= 5 \frac{π}{2} erf (\frac{x}{5})

Add a constant to the solution

= 5 \frac{π}{2} erf (\frac{x}{5}) + C

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What is the integral of e^{(-x^2)/5} ?
The integral of e^{(-x^2)/5} is sqrt(5)(sqrt(pi))/2 \erf(x/(sqrt(5)))+C