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Integral of exp(-2ax^2)
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Identical expressions
exp(- two ax^2)
exponent of ( minus 2ax squared )
exponent of ( minus two ax squared )
exp(-2ax2)
exp-2ax2
exp(-2ax²)
exp(-2ax to the power of 2)
exp-2ax^2
exp(-2ax^2)dx
Similar expressions
exp(2ax^2)

Integral of exp(-2ax^2) dx

The solution

You have entered [src]

  1            
  /            
 |             
 |         2   
 |   -2*a*x    
 |  e        dx
 |             
/              
0

$$\int\limits_{0}^{1} e^{- 2 a x^{2}}\, dx$$

Simplify

Detail solution

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

Now simplify:

$\frac{\sqrt{2} \sqrt{\pi} \sqrt{- \frac{1}{a}} \operatorname{erfi}{\left(\sqrt{2} x \sqrt{- a} \right)}}{4}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

                                      _____     /      ___\
  /                    ___   ____    / -1       |a*x*\/ 2 |
 |                   \/ 2 *\/ pi *  /  --- *erfi|---------|
 |        2                       \/    a       |    ____ |
 |  -2*a*x                                      \  \/ -a  /
 | e        dx = C - --------------------------------------
 |                                     4                   
/

$${{\sqrt{\pi}\,\mathrm{erf}\left(\sqrt{2}\,\sqrt{a}\,x\right)}\over{ 2^{{{3}\over{2}}}\,\sqrt{a}}}$$

Simplify

The answer [src]

/  ___   ____    /  ___   ___\                                  
|\/ 2 *\/ pi *erf\\/ 2 *\/ a /                                  
|-----------------------------  for And(a > -oo, a < oo, a != 0)
<               ___                                             
|           4*\/ a                                              
|                                                               
\              1                           otherwise

$${{\sqrt{\pi}\,\mathrm{erf}\left(\sqrt{2}\,\sqrt{a}\right)}\over{2^{ {{3}\over{2}}}\,\sqrt{a}}}$$

/  ___   ____    /  ___   ___\                                  
|\/ 2 *\/ pi *erf\\/ 2 *\/ a /                                  
|-----------------------------  for And(a > -oo, a < oo, a != 0)
<               ___                                             
|           4*\/ a                                              
|                                                               
\              1                           otherwise

$$\begin{cases} \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\sqrt{2} \sqrt{a} \right)}}{4 \sqrt{a}} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\1 & \text{otherwise} \end{cases}$$

Simplify

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

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Integral of exp(-2ax^2) dx

Limits of integration:

The graph:

Piecewise:

The solution