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Integral de (1/pi)2x*cos(ax) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
  0                 
  /                 
 |                  
 |  2               
 |  --*x*cos(a*x) dx
 |  pi              
 |                  
/                   
-pi                 
$$\int\limits_{- \pi}^{0} x \frac{2}{\pi} \cos{\left(a x \right)}\, dx$$
Integral(((2/pi)*x)*cos(a*x), (x, -pi, 0))
Respuesta (Indefinida) [src]
                            //           2                      \                             
                            ||          x                       |                             
                            ||          --             for a = 0|                             
                            ||          2                       |                             
                            ||                                  |                             
                            ||/-cos(a*x)                        |                             
                          2*|<|----------  for a != 0           |                             
                            ||<    a                            |                             
                            |||                                 |       //   x      for a = 0\
                            ||\    0       otherwise            |       ||                   |
  /                         ||-----------------------  otherwise|   2*x*|
            
$$\int x \frac{2}{\pi} \cos{\left(a x \right)}\, dx = C + \frac{2 x \left(\begin{cases} x & \text{for}\: a = 0 \\\frac{\sin{\left(a x \right)}}{a} & \text{otherwise} \end{cases}\right)}{\pi} - \frac{2 \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a = 0 \\\frac{\begin{cases} - \frac{\cos{\left(a x \right)}}{a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases}}{a} & \text{otherwise} \end{cases}\right)}{\pi}$$
Respuesta [src]
/    /cos(pi*a)   pi*sin(pi*a)\                                          
|  2*|--------- + ------------|                                          
|    |     2           a      |                                          
|    \    a                   /     2                                    
<- ---------------------------- + -----  for And(a > -oo, a < oo, a != 0)
|               pi                    2                                  
|                                 pi*a                                   
|                                                                        
\                 -pi                               otherwise            
$$\begin{cases} - \frac{2 \left(\frac{\pi \sin{\left(\pi a \right)}}{a} + \frac{\cos{\left(\pi a \right)}}{a^{2}}\right)}{\pi} + \frac{2}{\pi a^{2}} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\- \pi & \text{otherwise} \end{cases}$$
=
=
/    /cos(pi*a)   pi*sin(pi*a)\                                          
|  2*|--------- + ------------|                                          
|    |     2           a      |                                          
|    \    a                   /     2                                    
<- ---------------------------- + -----  for And(a > -oo, a < oo, a != 0)
|               pi                    2                                  
|                                 pi*a                                   
|                                                                        
\                 -pi                               otherwise            
$$\begin{cases} - \frac{2 \left(\frac{\pi \sin{\left(\pi a \right)}}{a} + \frac{\cos{\left(\pi a \right)}}{a^{2}}\right)}{\pi} + \frac{2}{\pi a^{2}} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\- \pi & \text{otherwise} \end{cases}$$
Piecewise((-2*(cos(pi*a)/a^2 + pi*sin(pi*a)/a)/pi + 2/(pi*a^2), (a > -oo)∧(a < oo)∧(Ne(a, 0))), (-pi, True))

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.