What is the integral from 0 to infinity of e^{-ax^2} ?

The integral from 0 to infinity of e^{-ax^2} is (sqrt(pi)-sqrt(pi)\erf(0*sqrt(a)))/(2sqrt(a))

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integral from 0 to infinity of e^{-ax^2}

\int_{0}^{\infty} e^{- a x^{2}} d x

\frac{π - π erf ( 0 \cdot a )}{2 a}

Solution steps

\int_{0}^{\infty} e^{- a x^{2}} d x

Compute the indefinite integral:

\int e^{- a x^{2}} d x = \frac{π}{2 a} erf (a x) + C

Compute the boundaries:

\int_{0}^{\infty} e^{- a x^{2}} d x = \frac{π}{2 a} - \frac{π}{2 a} erf (a \cdot 0)

= \frac{π}{2 a} - \frac{π}{2 a} erf (a \cdot 0)

Simplify

= \frac{π - π erf ( 0 \cdot a )}{2 a}

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What is the integral from 0 to infinity of e^{-ax^2} ?
The integral from 0 to infinity of e^{-ax^2} is (sqrt(pi)-sqrt(pi)\erf(0*sqrt(a)))/(2sqrt(a))