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Graphing y =:
x^2*exp((-x^2)/2)
Derivative of:
x^2*exp((-x^2)/2)
Identical expressions
x^ two *exp((-x^ two)/ two)
x squared multiply by exponent of (( minus x squared ) divide by 2)
x to the power of two multiply by exponent of (( minus x to the power of two) divide by two)
x2*exp((-x2)/2)
x2*exp-x2/2
x²*exp((-x²)/2)
x to the power of 2*exp((-x to the power of 2)/2)
x^2exp((-x^2)/2)
x2exp((-x2)/2)
x2exp-x2/2
x^2exp-x^2/2
x^2*exp((-x^2) divide by 2)
x^2*exp((-x^2)/2)dx
Similar expressions
x^2*exp(-x^2/2)
x^2*exp((x^2)/2)

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

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

The solution

You have entered [src]

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

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

Integral(x^2*exp((-x^2)/2), (x, 0, 1))

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 $\operatorname{dv}{\left(x \right)} = e^{- \frac{x^{2}}{2}}$ .

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:

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

                          /  ___\
            ___   ____    |\/ 2 |
          \/ 2 *\/ pi *erf|-----|
   -1/2                   \  2  /
- e     + -----------------------
                     2

$$- \frac{1}{e^{\frac{1}{2}}} + \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2}}{2} \right)}}{2}$$

                          /  ___\
            ___   ____    |\/ 2 |
          \/ 2 *\/ pi *erf|-----|
   -1/2                   \  2  /
- e     + -----------------------
                     2

$$- \frac{1}{e^{\frac{1}{2}}} + \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2}}{2} \right)}}{2}$$

-exp(-1/2) + sqrt(2)*sqrt(pi)*erf(sqrt(2)/2)/2

Numerical answer [src]

0.249093732179515

0.249093732179515

The graph

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution