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Identical expressions
exp^(-2x/a)
exponent of to the power of ( minus 2x divide by a)
exp(-2x/a)
exp-2x/a
exp^-2x/a
exp^(-2x divide by a)
exp^(-2x/a)dx
Similar expressions
exp^(2x/a)

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

The solution

You have entered [src]

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

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

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

Detail solution

Let $u = \frac{\left(-1\right) 2 x}{a}$ .

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

$\int \left(- \frac{a e^{u}}{2}\right)\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int e^{u}\, du = - \frac{a \int e^{u}\, du}{2}$
  1. The integral of the exponential function is itself.
    
    $\int e^{u}\, du = e^{u}$
  So, the result is: $- \frac{a e^{u}}{2}$
Now substitute $u$ back in:

$- \frac{a e^{\frac{\left(-1\right) 2 x}{a}}}{2}$
Now simplify:

$- \frac{a e^{- \frac{2 x}{a}}}{2}$
Add the constant of integration:

$- \frac{a e^{- \frac{2 x}{a}}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{a e^{- \frac{2 x}{a}}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

/     a                       pi
|     -        for |arg(a)| < --
|     2                       2 
|                               
| oo                            
|  /                            
| |                             
< |   -2*x                      
| |   ----                      
| |    a                        
| |  e     dx      otherwise    
| |                             
|/                              
|0                              
\

$$\begin{cases} \frac{a}{2} & \text{for}\: \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2} \\\int\limits_{0}^{\infty} e^{- \frac{2 x}{a}}\, dx & \text{otherwise} \end{cases}$$

/     a                       pi
|     -        for |arg(a)| < --
|     2                       2 
|                               
| oo                            
|  /                            
| |                             
< |   -2*x                      
| |   ----                      
| |    a                        
| |  e     dx      otherwise    
| |                             
|/                              
|0                              
\

$$\begin{cases} \frac{a}{2} & \text{for}\: \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2} \\\int\limits_{0}^{\infty} e^{- \frac{2 x}{a}}\, dx & \text{otherwise} \end{cases}$$

Piecewise((a/2, Abs(arg(a)) < pi/2), (Integral(exp(-2*x/a), (x, 0, oo)), True))

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution