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Limit of the function:
x^3*exp(x^2)
Identical expressions
x^ three *exp(x^ two)
x cubed multiply by exponent of (x squared )
x to the power of three multiply by exponent of (x to the power of two)
x3*exp(x2)
x3*expx2
x³*exp(x²)
x to the power of 3*exp(x to the power of 2)
x^3exp(x^2)
x3exp(x2)
x3expx2
x^3expx^2
x^3*exp(x^2)dx

Solving integrals/ x^3*exp(x^2)

Integral of x^3*exp(x^2) dx

The solution

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  1            
  /            
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 |      / 2\   
 |   3  \x /   
 |  x *e     dx
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/              
0

$$\int\limits_{0}^{1} x^{3} e^{x^{2}}\, dx$$

Integral(x^3*exp(x^2), (x, 0, 1))

Detail solution

Let $u = x^{2}$ .

Then let $du = 2 x dx$ and substitute $\frac{du}{2}$ :

$\int \frac{u e^{u}}{2}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int u e^{u}\, du = \frac{\int u e^{u}\, du}{2}$
  1. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(u \right)} = u$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
    
    Then $\operatorname{du}{\left(u \right)} = 1$ .
    
    To find $v{\left(u \right)}$ :
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    Now evaluate the sub-integral.
  2. The integral of the exponential function is itself.
    
    $\int e^{u}\, du = e^{u}$
  So, the result is: $\frac{u e^{u}}{2} - \frac{e^{u}}{2}$
Now substitute $u$ back in:

$\frac{x^{2} e^{x^{2}}}{2} - \frac{e^{x^{2}}}{2}$
Now simplify:

$\frac{\left(x^{2} - 1\right) e^{x^{2}}}{2}$
Add the constant of integration:

$\frac{\left(x^{2} - 1\right) e^{x^{2}}}{2}+ \mathrm{constant}$

The answer is:

$\frac{\left(x^{2} - 1\right) e^{x^{2}}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                  
 |                    / 2\       / 2\
 |     / 2\           \x /    2  \x /
 |  3  \x /          e       x *e    
 | x *e     dx = C - ----- + --------
 |                     2        2    
/

$$\int x^{3} e^{x^{2}}\, dx = C + \frac{x^{2} e^{x^{2}}}{2} - \frac{e^{x^{2}}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/2

$$\frac{1}{2}$$

1/2

$$\frac{1}{2}$$

1/2

Numerical answer [src]

0.5

0.5

The graph

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

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Identical expressions

Integral of x^3*exp(x^2) dx

Limits of integration:

The graph:

Piecewise:

The solution