Mister Exam

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

The solution

You have entered [src]

  1           
  /           
 |            
 |   -2*x     
 |   ----     
 |    a       
 |  e    *x dx
 |            
/             
0

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

Integral(exp((-2*x)/a)*x, (x, 0, 1))

Detail solution

Use integration by parts:

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

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

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

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:

$\int \left(- \frac{a e^{- \frac{2 x}{a}}}{2}\right)\, dx = - \frac{a \int e^{- \frac{2 x}{a}}\, dx}{2}$
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: $\frac{a^{2} e^{- \frac{2 x}{a}}}{4}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

                   -2 
                   ---
 2   /   2      \   a 
a    \- a  - 2*a/*e   
-- + -----------------
4            4

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

                   -2 
                   ---
 2   /   2      \   a 
a    \- a  - 2*a/*e   
-- + -----------------
4            4

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

a^2/4 + (-a^2 - 2*a)*exp(-2/a)/4

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 dx

Limits of integration:

The graph:

Piecewise:

The solution

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How to use it?

Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

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