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Graphing y =:
x^2*exp(x^2)
Identical expressions
x^ two *exp(x^ two)
x squared multiply by exponent of (x squared )
x to the power of two multiply by exponent of (x to the power of two)
x2*exp(x2)
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x to the power of 2*exp(x to the power of 2)
x^2exp(x^2)
x2exp(x2)
x2expx2
x^2expx^2
x^2*exp(x^2)dx

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

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

The solution

You have entered [src]

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

$$\int\limits_{0}^{1} x^{2} e^{x^{2}}\, dx$$

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

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:

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

      ____        
E   \/ pi *erfi(1)
- - --------------
2         4

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

      ____        
E   \/ pi *erfi(1)
- - --------------
2         4

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

E/2 - sqrt(pi)*erfi(1)/4

Numerical answer [src]

0.627815041275932

0.627815041275932

The graph

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

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Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution