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Solving integrals/ exp(-2ax^2)

Integral of exp(-2ax^2) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1            
  /            
 |             
 |         2   
 |   -2*a*x    
 |  e        dx
 |             
/              
0
∫01e−2ax2 dx\int\limits_{0}^{1} e^{- 2 a x^{2}}\, dx
Simplify
Detail solution

    ErfRule(a=-2*a, b=0, c=0, context=exp(-2*a*x**2), symbol=x)

  1. Now simplify:

    2π−1aerfi⁡(2x−a)4\frac{\sqrt{2} \sqrt{\pi} \sqrt{- \frac{1}{a}} \operatorname{erfi}{\left(\sqrt{2} x \sqrt{- a} \right)}}{4}

  2. Add the constant of integration:

    2π−1aerfi⁡(2x−a)4+constant\frac{\sqrt{2} \sqrt{\pi} \sqrt{- \frac{1}{a}} \operatorname{erfi}{\left(\sqrt{2} x \sqrt{- a} \right)}}{4}+ \mathrm{constant}

The answer is:

2π−1aerfi⁡(2x−a)4+constant\frac{\sqrt{2} \sqrt{\pi} \sqrt{- \frac{1}{a}} \operatorname{erfi}{\left(\sqrt{2} x \sqrt{- a} \right)}}{4}+ \mathrm{constant}

The answer (Indefinite) [src] 
                                      _____     /      ___\
  /                    ___   ____    / -1       |a*x*\/ 2 |
 |                   \/ 2 *\/ pi *  /  --- *erfi|---------|
 |        2                       \/    a       |    ____ |
 |  -2*a*x                                      \  \/ -a  /
 | e        dx = C - --------------------------------------
 |                                     4                   
/
π erf(2 a x)232 a{{\sqrt{\pi}\,\mathrm{erf}\left(\sqrt{2}\,\sqrt{a}\,x\right)}\over{ 2^{{{3}\over{2}}}\,\sqrt{a}}}
Simplify
The answer [src] 
/  ___   ____    /  ___   ___\                                  
|\/ 2 *\/ pi *erf\\/ 2 *\/ a /                                  
|-----------------------------  for And(a > -oo, a < oo, a != 0)
<               ___                                             
|           4*\/ a                                              
|                                                               
\              1                           otherwise
π erf(2 a)232 a{{\sqrt{\pi}\,\mathrm{erf}\left(\sqrt{2}\,\sqrt{a}\right)}\over{2^{ {{3}\over{2}}}\,\sqrt{a}}} 
=
= 
/  ___   ____    /  ___   ___\                                  
|\/ 2 *\/ pi *erf\\/ 2 *\/ a /                                  
|-----------------------------  for And(a > -oo, a < oo, a != 0)
<               ___                                             
|           4*\/ a                                              
|                                                               
\              1                           otherwise
{2πerf⁡(2a)4afor a>−∞∧a<∞∧a≠01otherwise\begin{cases} \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\sqrt{2} \sqrt{a} \right)}}{4 \sqrt{a}} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\1 & \text{otherwise} \end{cases}
Simplify

    Use the examples entering the upper and lower limits of integration.

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