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exp(2x/a)*x^2

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

The solution

You have entered [src]

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

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

Integral(exp((-2*x)/a)*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 $\operatorname{dv}{\left(x \right)} = e^{- \frac{2 x}{a}}$ .

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

To find $v{\left(x \right)}$ :
1. Let $u = - \frac{2 x}{a}$ .
  
  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:
  
  $- \frac{a e^{- \frac{2 x}{a}}}{2}$
Now evaluate the sub-integral.
Use integration by parts:

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

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

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

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

$\int \frac{a^{2} e^{- \frac{2 x}{a}}}{2}\, dx = \frac{a^{2} \int e^{- \frac{2 x}{a}}\, dx}{2}$
1. Let $u = - \frac{2 x}{a}$ .
  
  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:
  
  $- \frac{a e^{- \frac{2 x}{a}}}{2}$
So, the result is: $- \frac{a^{3} e^{- \frac{2 x}{a}}}{4}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                    
 |                       -2*x         -2*x         -2*x
 |  -2*x                 ----         ----         ----
 |  ----              3   a        2   a        2   a  
 |   a    2          a *e       a*x *e       x*a *e    
 | e    *x  dx = C - -------- - ---------- - ----------
 |                      4           2            2     
/

$$\int x^{2} e^{\frac{\left(-1\right) 2 x}{a}}\, dx = C - \frac{a^{3} e^{- \frac{2 x}{a}}}{4} - \frac{a^{2} x e^{- \frac{2 x}{a}}}{2} - \frac{a x^{2} e^{- \frac{2 x}{a}}}{2}$$

Simplify

The answer [src]

/       3                          
|      a                         pi
|      --         for |arg(a)| < --
|      4                         2 
|                                  
| oo                               
|  /                               
< |                                
| |      -2*x                      
| |      ----                      
| |   2   a                        
| |  x *e     dx      otherwise    
| |                                
|/                                 
\0

$$\begin{cases} \frac{a^{3}}{4} & \text{for}\: \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2} \\\int\limits_{0}^{\infty} x^{2} e^{- \frac{2 x}{a}}\, dx & \text{otherwise} \end{cases}$$

/       3                          
|      a                         pi
|      --         for |arg(a)| < --
|      4                         2 
|                                  
| oo                               
|  /                               
< |                                
| |      -2*x                      
| |      ----                      
| |   2   a                        
| |  x *e     dx      otherwise    
| |                                
|/                                 
\0

$$\begin{cases} \frac{a^{3}}{4} & \text{for}\: \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2} \\\int\limits_{0}^{\infty} x^{2} e^{- \frac{2 x}{a}}\, dx & \text{otherwise} \end{cases}$$

Piecewise((a^3/4, Abs(arg(a)) < pi/2), (Integral(x^2*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)*x^2 dx

Limits of integration:

The graph:

Piecewise:

The solution