What is the integral of e^{-a*x^2} ?

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

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integral of e^{-a*x^2}

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

\frac{π}{2 a} erf (a x) + C

Solution steps

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

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

Apply u-substitution

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

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

Take the constant out:

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

= \frac{1}{a} \cdot \int e^{- u^{2}} d u

Use the nonelementary integral:

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

= \frac{1}{a} \cdot \frac{π}{2} erf (u)

Substitute back

u = a x

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

Simplify

\frac{1}{a} \cdot \frac{π}{2} erf (a x) : \frac{π}{2 a} erf (a x)

= \frac{π}{2 a} erf (a x)

Add a constant to the solution

= \frac{π}{2 a} erf (a x) + C

f (x) = 5 cos (x)

\frac{d}{d x} (\frac{0.25 e ^{x}}{1 - 0.75 e ^{x}})

y^{^{''}} - 2 y^{^{'}} - 3 y = - 6 cos (t) - 9 sin (t)

\int 3 x^{2} + 5 x^{\frac{1}{3}} + 7 sin (x) d x

sin (x), cos (x), [0, π]

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