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Integral of d{x}:
Integral of x²
Integral of sin
Integral of sin^2x
Integral of 2/x^2
Identical expressions
x^ two e^(-x^2)
x squared e to the power of ( minus x squared )
x to the power of two e to the power of ( minus x squared )
x2e(-x2)
x2e-x2
x²e^(-x²)
x to the power of 2e to the power of (-x to the power of 2)
x^2e^-x^2
x^2e^(-x^2)dx
Similar expressions
x^2e^(x^2)
What you mean?
x^2/e^(x^2)
x^((2*e)^(-x^2))
x^((2*e)^(-x^2))

You entered:

x^2e^(-x^2)

What you mean?

x^2/e^(x^2)

x^((2*e)^(-x^2))

x^((2*e)^(-x^2))

Integral of x^2e^(-x^2) dx

The solution

You have entered [src]

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

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

Integral(x^2/E^(x^2), (x, exp(-1), E))

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

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:

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

$${{\sqrt{\pi}\,\mathrm{erf}\left(x\right)}\over{4}}-{{x\,e^ {- x^2 } }\over{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       -2        2                                  
 -1  -e        -e      ____    / -1\     ____       
e  *e       e*e      \/ pi *erf\e  /   \/ pi *erf(e)
--------- - ------ - --------------- + -------------
    2         2             4                4

$${{e^ {- E^2 }\,\left(\sqrt{\pi}\,e^{E^2}\,\mathrm{erf}\left(E \right)-2\,E\right)}\over{4}}-{{e^{-e^ {- 2 }-1}\,\left(\sqrt{\pi}\, e^{e^ {- 2 }+1}\,\mathrm{erf}\left(e^ {- 1 }\right)-2\right)}\over{4 }}$$

       -2        2                                  
 -1  -e        -e      ____    / -1\     ____       
e  *e       e*e      \/ pi *erf\e  /   \/ pi *erf(e)
--------- - ------ - --------------- + -------------
    2         2             4                4

$$- \frac{\sqrt{\pi} \operatorname{erf}{\left(e^{-1} \right)}}{4} - \frac{e}{2 e^{e^{2}}} + \frac{1}{2 e e^{e^{-2}}} + \frac{\sqrt{\pi} \operatorname{erf}{\left(e \right)}}{4}$$

Упростить

Numerical answer [src]

0.426908945067475

0.426908945067475

The graph

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

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What you mean?

Integral of x^2e^(-x^2) dx

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The solution