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Identical expressions
zero .25xe^(- zero .5x)
0.25xe to the power of ( minus 0.5x)
zero .25xe to the power of ( minus zero .5x)
0.25xe(-0.5x)
0.25xe-0.5x
0.25xe^-0.5x
0.25xe^(-0.5x)dx
Similar expressions
0.25xe^(0.5x)

Integral of 0.25xe^(-0.5x) dx

The solution

You have entered [src]

 oo          
  /          
 |           
 |     -x    
 |     ---   
 |  x   2    
 |  -*E    dx
 |  4        
 |           
/            
0

$$\int\limits_{0}^{\infty} e^{- \frac{x}{2}} \frac{x}{4}\, dx$$

Integral((x/4)*E^(-x/2), (x, 0, oo))

Detail solution

Let $u = - \frac{x}{2}$ .

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

$\int u e^{u}\, du$
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.
  
  $\int e^{u}\, du = e^{u}$
Now substitute $u$ back in:

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

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                             
 |                           -x 
 |    -x            -x       ---
 |    ---           ---       2 
 | x   2             2    x*e   
 | -*E    dx = C - e    - ------
 | 4                        2   
 |                              
/

$$\int e^{- \frac{x}{2}} \frac{x}{4}\, dx = C - \frac{x e^{- \frac{x}{2}}}{2} - e^{- \frac{x}{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$1$$

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

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Identical expressions

Similar expressions

Integral of 0.25xe^(-0.5x) dx

Limits of integration:

The graph:

Piecewise:

The solution