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Integral de (1-x)(cos(n*x)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
  0                    
  /                    
 |                     
 |  (1 - x)*cos(n*x) dx
 |                     
/                      
-pi                    
$$\int\limits_{- \pi}^{0} \left(1 - x\right) \cos{\left(n x \right)}\, dx$$
Integral((1 - x)*cos(n*x), (x, -pi, 0))
Respuesta (Indefinida) [src]
                                                                                 //           2                      \
                                                                                 ||          x                       |
                                                                                 ||          --             for n = 0|
                                                                                 ||          2                       |
  /                            //   x      for n = 0\   //   x      for n = 0\   ||                                  |
 |                             ||                   |   ||                   |   ||/-cos(n*x)                        |
 | (1 - x)*cos(n*x) dx = C - x*|
            
$$\int \left(1 - x\right) \cos{\left(n x \right)}\, dx = C - x \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right) + \begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(n x \right)}}{n} & \text{otherwise} \end{cases} + \begin{cases} \frac{x^{2}}{2} & \text{for}\: n = 0 \\\frac{\begin{cases} - \frac{\cos{\left(n x \right)}}{n} & \text{for}\: n \neq 0 \\0 & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases}$$
Respuesta [src]
/  1    sin(pi*n)   cos(pi*n)   pi*sin(pi*n)                                  
|- -- + --------- + --------- + ------------  for And(n > -oo, n < oo, n != 0)
|   2       n            2           n                                        
|  n                    n                                                     
<                                                                             
|                        2                                                    
|                      pi                                                     
|                 pi + ---                               otherwise            
\                       2                                                     
$$\begin{cases} \frac{\sin{\left(\pi n \right)}}{n} + \frac{\pi \sin{\left(\pi n \right)}}{n} + \frac{\cos{\left(\pi n \right)}}{n^{2}} - \frac{1}{n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\pi + \frac{\pi^{2}}{2} & \text{otherwise} \end{cases}$$
=
=
/  1    sin(pi*n)   cos(pi*n)   pi*sin(pi*n)                                  
|- -- + --------- + --------- + ------------  for And(n > -oo, n < oo, n != 0)
|   2       n            2           n                                        
|  n                    n                                                     
<                                                                             
|                        2                                                    
|                      pi                                                     
|                 pi + ---                               otherwise            
\                       2                                                     
$$\begin{cases} \frac{\sin{\left(\pi n \right)}}{n} + \frac{\pi \sin{\left(\pi n \right)}}{n} + \frac{\cos{\left(\pi n \right)}}{n^{2}} - \frac{1}{n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\pi + \frac{\pi^{2}}{2} & \text{otherwise} \end{cases}$$
Piecewise((-1/n^2 + sin(pi*n)/n + cos(pi*n)/n^2 + pi*sin(pi*n)/n, (n > -oo)∧(n < oo)∧(Ne(n, 0))), (pi + pi^2/2, True))

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