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Solving integrals/ exp(-(x^2)/2)/2/pi

Integral of exp(-(x^2)/2)/2/pi dx

The solution

You have entered [src]

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

$$\int\limits_{0}^{1} \frac{e^{\frac{\left(-1\right) x^{2}}{2}}}{2 \pi}\, dx$$

Simplify

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{e^{\frac{\left(-1\right) x^{2}}{2}}}{2 \pi}\, dx = \frac{\int e^{\frac{\left(-1\right) x^{2}}{2}}\, dx}{2 \pi}$
So, the result is: $\frac{\sqrt{2} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{4 \sqrt{\pi}}$
Add the constant of integration:

$\frac{\sqrt{2} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{4 \sqrt{\pi}}+ \mathrm{constant}$

The answer is:

$\frac{\sqrt{2} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{4 \sqrt{\pi}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                 
 |                                  
 |    2                             
 |  -x                     /    ___\
 |  ----            ___    |x*\/ 2 |
 |   2            \/ 2 *erf|-------|
 | e                       \   2   /
 | ----- dx = C + ------------------
 |  2*pi                   ____     
 |                     4*\/ pi      
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

         /  ___\
  ___    |\/ 2 |
\/ 2 *erf|-----|
         \  2  /
----------------
        ____    
    4*\/ pi

$${{\sqrt{\pi}\,\mathrm{erf}\left({{1}\over{\sqrt{2}}}\right)}\over{2 ^{{{3}\over{2}}}\,\pi}}$$

         /  ___\
  ___    |\/ 2 |
\/ 2 *erf|-----|
         \  2  /
----------------
        ____    
    4*\/ pi

$$\frac{\sqrt{2} \operatorname{erf}{\left(\frac{\sqrt{2}}{2} \right)}}{4 \sqrt{\pi}}$$

Simplify

Numerical answer [src]

0.136176851399633

0.136176851399633

The graph

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

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Identical expressions

Similar expressions

Integral of exp(-(x^2)/2)/2/pi dx

Limits of integration:

The graph:

Piecewise:

The solution