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Integral of t*exp(-t/2) dt

The solution

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 oo          
  /          
 |           
 |     -t    
 |     ---   
 |      2    
 |  t*e    dt
 |           
/            
0

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

Integral(t*exp((-t)/2), (t, 0, oo))

Detail solution

Use integration by parts:

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

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

Then $\operatorname{du}{\left(t \right)} = 1$ .

To find $v{\left(t \right)}$ :
1. Let $u = - \frac{t}{2}$ .
  
  Then let $du = - \frac{dt}{2}$ 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:
  
  $- 2 e^{- \frac{t}{2}}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- 2 e^{- \frac{t}{2}}\right)\, dt = - 2 \int e^{- \frac{t}{2}}\, dt$
1. Let $u = - \frac{t}{2}$ .
  
  Then let $du = - \frac{dt}{2}$ 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:
  
  $- 2 e^{- \frac{t}{2}}$
So, the result is: $4 e^{- \frac{t}{2}}$
Now simplify:

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

$- \left(2 t + 4\right) e^{- \frac{t}{2}}+ \mathrm{constant}$

The answer is:

$- \left(2 t + 4\right) e^{- \frac{t}{2}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                 
 |                                  
 |    -t              -t         -t 
 |    ---             ---        ---
 |     2               2          2 
 | t*e    dt = C - 4*e    - 2*t*e   
 |                                  
/

$$\int t e^{\frac{\left(-1\right) t}{2}}\, dt = C - 2 t e^{- \frac{t}{2}} - 4 e^{- \frac{t}{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$4$$

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

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Identical expressions

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Integral of t*exp(-t/2) dt

Limits of integration:

The graph:

Piecewise:

The solution