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  • x^2*exp(-a*(x^2)/2)dx

  • Similar expressions

  • x^2*exp(a*(x^2)/2)
Solving integrals/ x^2*exp(-a*(x^2)/2)

Integral of x^2*exp(-a*(x^2)/2) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1             
  /             
 |              
 |          2   
 |      -a*x    
 |      -----   
 |   2    2     
 |  x *e      dx
 |              
/               
0
$$\int\limits_{0}^{1} x^{2} e^{\frac{- a x^{2}}{2}}\, dx$$ 
Integral(x^2*exp(((-a)*x^2)/2), (x, 0, 1))
Detail solution

  1. Use integration by parts:

    Let and let .

    Then .

    To find :

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

    Now evaluate the sub-integral.

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

    1. Don't know the steps in finding this integral.

      But the integral is

    So, the result is:

  3. Now simplify:

  4. Add the constant of integration:

The answer is:

The answer (Indefinite) [src] 
  /                                                                                                                                                            
 |                                           /                                                                  2 \                       _____     /      ___\
 |         2                                 |    /      ___   ___\          /      ___   ___\              -a*x  |     ___   ____  2    / -1       |a*x*\/ 2 |
 |     -a*x                                  |    |I*x*\/ 2 *\/ a |    2     |I*x*\/ 2 *\/ a |              ------|   \/ 2 *\/ pi *x *  /  --- *erfi|---------|
 |     -----                           _____ |erfi|---------------|   x *erfi|---------------|         ___    2   |                   \/    a       |     ____|
 |  2    2              ___   ____    / -1   |    \       2       /          \       2       /   I*x*\/ 2 *e      |                                 \ 2*\/ -a /
 | x *e      dx = C + \/ 2 *\/ pi *  /  --- *|--------------------- - ------------------------ - -----------------| - -----------------------------------------
 |                                 \/    a   |         2*a                       2                     ____   ___ |                       2                    
/                                            \                                                     2*\/ pi *\/ a  /
$$\int x^{2} e^{\frac{- a x^{2}}{2}}\, dx = C - \frac{\sqrt{2} \sqrt{\pi} x^{2} \sqrt{- \frac{1}{a}} \operatorname{erfi}{\left(\frac{\sqrt{2} a x}{2 \sqrt{- a}} \right)}}{2} + \sqrt{2} \sqrt{\pi} \sqrt{- \frac{1}{a}} \left(- \frac{x^{2} \operatorname{erfi}{\left(\frac{\sqrt{2} i \sqrt{a} x}{2} \right)}}{2} + \frac{\operatorname{erfi}{\left(\frac{\sqrt{2} i \sqrt{a} x}{2} \right)}}{2 a} - \frac{\sqrt{2} i x e^{- \frac{a x^{2}}{2}}}{2 \sqrt{\pi} \sqrt{a}}\right)$$
Simplify
The answer [src] 
/   -a                    /  ___   ___\                                  
|   ---     ___   ____    |\/ 2 *\/ a |                                  
|    2    \/ 2 *\/ pi *erf|-----------|                                  
|  e                      \     2     /                                  
<- ---- + -----------------------------  for And(a > -oo, a < oo, a != 0)
|   a                    3/2                                             
|                     2*a                                                
|                                                                        
\                 1/3                               otherwise
$$\begin{cases} - \frac{e^{- \frac{a}{2}}}{a} + \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} \sqrt{a}}{2} \right)}}{2 a^{\frac{3}{2}}} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\\frac{1}{3} & \text{otherwise} \end{cases}$$ 
=
= 
/   -a                    /  ___   ___\                                  
|   ---     ___   ____    |\/ 2 *\/ a |                                  
|    2    \/ 2 *\/ pi *erf|-----------|                                  
|  e                      \     2     /                                  
<- ---- + -----------------------------  for And(a > -oo, a < oo, a != 0)
|   a                    3/2                                             
|                     2*a                                                
|                                                                        
\                 1/3                               otherwise
$$\begin{cases} - \frac{e^{- \frac{a}{2}}}{a} + \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} \sqrt{a}}{2} \right)}}{2 a^{\frac{3}{2}}} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\\frac{1}{3} & \text{otherwise} \end{cases}$$ 
Piecewise((-exp(-a/2)/a + sqrt(2)*sqrt(pi)*erf(sqrt(2)*sqrt(a)/2)/(2*a^(3/2)), (a > -oo)∧(a < oo)∧(Ne(a, 0))), (1/3, True))

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

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