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x^2exp(ax^2)

Integral of x^2exp(-ax^2) dx

The solution

You have entered [src]

 oo             
  /             
 |              
 |          2   
 |   2  -a*x    
 |  x *e      dx
 |              
/               
-oo

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

Integral(x^2*exp((-a)*x^2), (x, -oo, oo))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = x^{2}$ and let $\operatorname{dv}{\left(x \right)} = e^{- a x^{2}}$ .

Then $\operatorname{du}{\left(x \right)} = 2 x$ .

To find $v{\left(x \right)}$ :
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \sqrt{\pi} x \sqrt{- \frac{1}{a}} \operatorname{erfi}{\left(\frac{a x}{\sqrt{- a}} \right)}\right)\, dx = - \sqrt{\pi} \sqrt{- \frac{1}{a}} \int x \operatorname{erfi}{\left(\frac{a x}{\sqrt{- a}} \right)}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $- \frac{x^{2} \operatorname{erfi}{\left(i \sqrt{a} x \right)}}{2} + \frac{\operatorname{erfi}{\left(i \sqrt{a} x \right)}}{4 a} - \frac{i x e^{- a x^{2}}}{2 \sqrt{\pi} \sqrt{a}}$
So, the result is: $- \sqrt{\pi} \sqrt{- \frac{1}{a}} \left(- \frac{x^{2} \operatorname{erfi}{\left(i \sqrt{a} x \right)}}{2} + \frac{\operatorname{erfi}{\left(i \sqrt{a} x \right)}}{4 a} - \frac{i x e^{- a x^{2}}}{2 \sqrt{\pi} \sqrt{a}}\right)$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

$$\int x^{2} e^{- a x^{2}}\, dx = C - \frac{\sqrt{\pi} x^{2} \sqrt{- \frac{1}{a}} \operatorname{erfi}{\left(\frac{a x}{\sqrt{- a}} \right)}}{2} + \sqrt{\pi} \sqrt{- \frac{1}{a}} \left(- \frac{x^{2} \operatorname{erfi}{\left(i \sqrt{a} x \right)}}{2} + \frac{\operatorname{erfi}{\left(i \sqrt{a} x \right)}}{4 a} - \frac{i x e^{- a x^{2}}}{2 \sqrt{\pi} \sqrt{a}}\right)$$

Simplify

The answer [src]

/       ____                        
|     \/ pi                       pi
|     ------       for |arg(a)| < --
|        3/2                      2 
|     2*a                           
|                                   
| oo                                
<  /                                
| |                                 
| |          2                      
| |   2  -a*x                       
| |  x *e      dx      otherwise    
| |                                 
|/                                  
\-oo

$$\begin{cases} \frac{\sqrt{\pi}}{2 a^{\frac{3}{2}}} & \text{for}\: \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2} \\\int\limits_{-\infty}^{\infty} x^{2} e^{- a x^{2}}\, dx & \text{otherwise} \end{cases}$$

/       ____                        
|     \/ pi                       pi
|     ------       for |arg(a)| < --
|        3/2                      2 
|     2*a                           
|                                   
| oo                                
<  /                                
| |                                 
| |          2                      
| |   2  -a*x                       
| |  x *e      dx      otherwise    
| |                                 
|/                                  
\-oo

Piecewise((sqrt(pi)/(2*a^(3/2)), Abs(arg(a)) < pi/2), (Integral(x^2*exp(-a*x^2), (x, -oo, oo)), True))

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

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Identical expressions

Similar expressions

Integral of x^2exp(-ax^2) dx

Limits of integration:

The graph:

Piecewise:

The solution