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Integral of d{x}:
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Integral of x*dx/(x^2+1)
Identical expressions
x^ three *e^(x/ three)
x cubed multiply by e to the power of (x divide by 3)
x to the power of three multiply by e to the power of (x divide by three)
x3*e(x/3)
x3*ex/3
x³*e^(x/3)
x to the power of 3*e to the power of (x/3)
x^3e^(x/3)
x3e(x/3)
x3ex/3
x^3e^x/3
x^3*e^(x divide by 3)
x^3*e^(x/3)dx
What you mean?
x^3*e^(x/3)
x^((3*e)^(x/3))
x^((3*e)^(x/3))

You entered:

x^3*e^(x/3)

What you mean?

x^3*e^(x/3)

x^((3*e)^(x/3))

x^((3*e)^(x/3))

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

The solution

You have entered [src]

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

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

Integral(x^3*E^(x/3), (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 $\operatorname{dv}{\left(x \right)} = e^{\frac{x}{3}}$ .

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 $u = \frac{x}{3}$ .
    
    Then let $du = \frac{dx}{3}$ and substitute $3 du$ :
    
    $\int 9 e^{u}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 3 e^{u}\, du = 3 \int e^{u}\, du$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $3 e^{u}$
    Now substitute $u$ back in:
    
    $3 e^{\frac{x}{3}}$
  Method #2
  1. Let $u = e^{\frac{x}{3}}$ .
    
    Then let $du = \frac{e^{\frac{x}{3}} dx}{3}$ and substitute $3 du$ :
    
    $\int 9\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 3\, du = 3 \int 1\, du$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $3 u$
    Now substitute $u$ back in:
    
    $3 e^{\frac{x}{3}}$
Now evaluate the sub-integral.
Use integration by parts:

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

Let $u{\left(x \right)} = 9 x^{2}$ and let $\operatorname{dv}{\left(x \right)} = e^{\frac{x}{3}}$ .

Then $\operatorname{du}{\left(x \right)} = 18 x$ .

To find $v{\left(x \right)}$ :
1. Let $u = \frac{x}{3}$ .
  
  Then let $du = \frac{dx}{3}$ and substitute $3 du$ :
  
  $\int 9 e^{u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 e^{u}\, du = 3 \int e^{u}\, du$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $3 e^{u}$
  Now substitute $u$ back in:
  
  $3 e^{\frac{x}{3}}$
Now evaluate the sub-integral.
Use integration by parts:

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

Let $u{\left(x \right)} = 54 x$ and let $\operatorname{dv}{\left(x \right)} = e^{\frac{x}{3}}$ .

Then $\operatorname{du}{\left(x \right)} = 54$ .

To find $v{\left(x \right)}$ :
1. Let $u = \frac{x}{3}$ .
  
  Then let $du = \frac{dx}{3}$ and substitute $3 du$ :
  
  $\int 9 e^{u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 e^{u}\, du = 3 \int e^{u}\, du$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $3 e^{u}$
  Now substitute $u$ back in:
  
  $3 e^{\frac{x}{3}}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int 162 e^{\frac{x}{3}}\, dx = 162 \int e^{\frac{x}{3}}\, dx$
1. Let $u = \frac{x}{3}$ .
  
  Then let $du = \frac{dx}{3}$ and substitute $3 du$ :
  
  $\int 9 e^{u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 e^{u}\, du = 3 \int e^{u}\, du$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $3 e^{u}$
  Now substitute $u$ back in:
  
  $3 e^{\frac{x}{3}}$
So, the result is: $486 e^{\frac{x}{3}}$
Now simplify:

$3 \left(x^{3} - 9 x^{2} + 54 x - 162\right) e^{\frac{x}{3}}$
Add the constant of integration:

$3 \left(x^{3} - 9 x^{2} + 54 x - 162\right) e^{\frac{x}{3}}+ \mathrm{constant}$

The answer is:

$3 \left(x^{3} - 9 x^{2} + 54 x - 162\right) e^{\frac{x}{3}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                     
 |                                                      
 |     x               x          x         x          x
 |     -               -          -         -          -
 |  3  3               3       2  3      3  3          3
 | x *e  dx = C - 486*e  - 27*x *e  + 3*x *e  + 162*x*e 
 |                                                      
/

$$\left(3\,x^3-27\,x^2+162\,x-486\right)\,e^{{{x}\over{3}}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

           1/3
486 - 348*e

$$486-348\,e^{{{1}\over{3}}}$$

           1/3
486 - 348*e

$$486 - 348 e^{\frac{1}{3}}$$

Упростить

Numerical answer [src]

0.326876070040844

0.326876070040844

The graph

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

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Identical expressions

What you mean?

You entered:

What you mean?

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2