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Integral of e^(x^2)*x^3
Limit of the function:
e^(x^2)*x^3
Identical expressions
e^(x^ two)*x^ three
e to the power of (x squared ) multiply by x cubed
e to the power of (x to the power of two) multiply by x to the power of three
e(x2)*x3
ex2*x3
e^(x²)*x³
e to the power of (x to the power of 2)*x to the power of 3
e^(x^2)x^3
e(x2)x3
ex2x3
e^x^2x^3
e^(x^2)*x^3dx

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

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

The solution

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

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

Integral(E^(x^2)*x^3, (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 /
 |  \x /  3          e       x *e    
 | E    *x  dx = C - ----- + --------
 |                     2        2    
/

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

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 e^(x^2)*x^3 dx

Limits of integration:

The graph:

Piecewise:

The solution