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x^3e^2xdx
What you mean?
x^3*e^(2*x)*dx
x^((3*e)^(2*x))*dx
x^((3*e)^(2*x))*dx

You entered:

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

What you mean?

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

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

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

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

The solution

You have entered [src]

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

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

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

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = x^{3}$ and let $\operatorname{dv}{\left(x \right)} = e^{2 x}$ .

Then $\operatorname{du}{\left(x \right)} = 3 x^{2}$ .

To find $v{\left(x \right)}$ :
1. There are multiple ways to do this integral.
  Method #1
  1. Let $u = 2 x$ .
    
    Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{e^{u}}{4}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{u}}{2}\, du = \frac{\int e^{u}\, du}{2}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{2}$
    Now substitute $u$ back in:
    
    $\frac{e^{2 x}}{2}$
  Method #2
  1. Let $u = e^{2 x}$ .
    
    Then let $du = 2 e^{2 x} dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{1}{4}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{2}\, du = \frac{\int 1\, du}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $\frac{u}{2}$
    Now substitute $u$ back in:
    
    $\frac{e^{2 x}}{2}$
Now evaluate the sub-integral.
Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = \frac{3 x^{2}}{2}$ and let $\operatorname{dv}{\left(x \right)} = e^{2 x}$ .

Then $\operatorname{du}{\left(x \right)} = 3 x$ .

To find $v{\left(x \right)}$ :
1. Let $u = 2 x$ .
  
  Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{e^{u}}{4}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{e^{u}}{2}\, du = \frac{\int e^{u}\, du}{2}$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $\frac{e^{u}}{2}$
  Now substitute $u$ back in:
  
  $\frac{e^{2 x}}{2}$
Now evaluate the sub-integral.
Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = \frac{3 x}{2}$ and let $\operatorname{dv}{\left(x \right)} = e^{2 x}$ .

Then $\operatorname{du}{\left(x \right)} = \frac{3}{2}$ .

To find $v{\left(x \right)}$ :
1. Let $u = 2 x$ .
  
  Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{e^{u}}{4}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{e^{u}}{2}\, du = \frac{\int e^{u}\, du}{2}$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $\frac{e^{u}}{2}$
  Now substitute $u$ back in:
  
  $\frac{e^{2 x}}{2}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{3 e^{2 x}}{4}\, dx = \frac{3 \int e^{2 x}\, dx}{4}$
1. Let $u = 2 x$ .
  
  Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{e^{u}}{4}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{e^{u}}{2}\, du = \frac{\int e^{u}\, du}{2}$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $\frac{e^{u}}{2}$
  Now substitute $u$ back in:
  
  $\frac{e^{2 x}}{2}$
So, the result is: $\frac{3 e^{2 x}}{8}$
Now simplify:

$\frac{\left(4 x^{3} - 6 x^{2} + 6 x - 3\right) e^{2 x}}{8}$
Add the constant of integration:

$\frac{\left(4 x^{3} - 6 x^{2} + 6 x - 3\right) e^{2 x}}{8}+ \mathrm{constant}$

The answer is:

$\frac{\left(4 x^{3} - 6 x^{2} + 6 x - 3\right) e^{2 x}}{8}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                          
 |                       2*x    3  2*x      2  2*x        2*x
 |  3  2*x            3*e      x *e      3*x *e      3*x*e   
 | x *e   *1 dx = C - ------ + ------- - --------- + --------
 |                      8         2          4          4    
/

$${{\left(4\,x^3-6\,x^2+6\,x-3\right)\,e^{2\,x}}\over{8}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$${{e^2}\over{8}}+{{3}\over{8}}$$

$$\frac{3}{8} + \frac{e^{2}}{8}$$

Упростить

Numerical answer [src]

1.29863201236633

1.29863201236633

The graph

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

Other calculators

How to use it?

Identical expressions

What you mean?

You entered:

What you mean?

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Other calculators

How to use it?

Identical expressions

What you mean?

You entered:

What you mean?

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

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