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

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

The solution

You have entered [src]

 oo           
  /           
 |            
 |      -x    
 |      ---   
 |   2   2    
 |  x *e    dx
 |            
/             
0

$$\int\limits_{0}^{\infty} x^{2} e^{\frac{\left(-1\right) x}{2}}\, dx$$

Integral(x^2*exp((-x)/2), (x, 0, oo))

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 $\operatorname{dv}{\left(x \right)} = e^{- \frac{x}{2}}$ .

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

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

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

Let $u{\left(x \right)} = - 4 x$ and let $\operatorname{dv}{\left(x \right)} = e^{- \frac{x}{2}}$ .

Then $\operatorname{du}{\left(x \right)} = -4$ .

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

$\int 8 e^{- \frac{x}{2}}\, dx = 8 \int e^{- \frac{x}{2}}\, dx$
1. Let $u = - \frac{x}{2}$ .
  
  Then let $du = - \frac{dx}{2}$ and substitute $- 2 du$ :
  
  $\int \left(- 2 e^{u}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\text{False}$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $- 2 e^{u}$
  Now substitute $u$ back in:
  
  $- 2 e^{- \frac{x}{2}}$
So, the result is: $- 16 e^{- \frac{x}{2}}$
Now simplify:

$- \left(2 x^{2} + 8 x + 16\right) e^{- \frac{x}{2}}$
Add the constant of integration:

$- \left(2 x^{2} + 8 x + 16\right) e^{- \frac{x}{2}}+ \mathrm{constant}$

The answer is:

$- \left(2 x^{2} + 8 x + 16\right) e^{- \frac{x}{2}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                               
 |                                                
 |     -x               -x         -x          -x 
 |     ---              ---        ---         ---
 |  2   2                2          2       2   2 
 | x *e    dx = C - 16*e    - 8*x*e    - 2*x *e   
 |                                                
/

$$\int x^{2} e^{\frac{\left(-1\right) x}{2}}\, dx = C - 2 x^{2} e^{- \frac{x}{2}} - 8 x e^{- \frac{x}{2}} - 16 e^{- \frac{x}{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$16$$

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

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

Limits of integration:

The graph:

Piecewise:

The solution