What is the integral of e^{(-x^2)/3} ?

The integral of e^{(-x^2)/3} is sqrt(3)(sqrt(pi))/2 \erf(x/(sqrt(3)))+C

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integral of e^{(-x^2)/3}

integral

e^{\frac{- x ^{2}}{3}}

3 \frac{π}{2} erf (\frac{x}{3}) + C

Solution steps

\int e^{\frac{- x ^{2}}{3}} d x

Simplify

= \int e^{- \frac{x ^{2}}{3}} d x

Apply exponent rule:

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

\frac{x ^{2}}{3} = (\frac{x}{3})^{2}

= \int e^{- (\frac{x}{3})^{2}} d x

Apply u-substitution

: \int e^{- u^{2}} 3 d u

= \int e^{- u^{2}} 3 d u

Take the constant out:

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

= 3 \cdot \int e^{- u^{2}} d u

Use the nonelementary integral:

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

= 3 \frac{π}{2} erf (u)

Substitute back

u = \frac{x}{3}

= 3 \frac{π}{2} erf (\frac{x}{3})

Add a constant to the solution

= 3 \frac{π}{2} erf (\frac{x}{3}) + C

\frac{( x + 2 )}{( x ^{2} + 1 )}

(1 + t^{2})^{- 1}

\frac{( 7 ^{x} - 1 )}{x}

\frac{x}{4 + x}

\frac{1}{( x ^{2} + 2 x + 4 ) ^{\frac{5}{2}}}

What is the integral of e^{(-x^2)/3} ?
The integral of e^{(-x^2)/3} is sqrt(3)(sqrt(pi))/2 \erf(x/(sqrt(3)))+C