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Resolución de integrales/ cos(n*x)/ (3*x-1)*cos(n*x)

Integral de (3*x-1)*cos(n*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 pi                      
  /                      
 |                       
 |  (3*x - 1)*cos(n*x) dx
 |                       
/                        
0
$$\int\limits_{0}^{\pi} \left(3 x - 1\right) \cos{\left(n x \right)}\, dx$$ 
Integral((3*x - 1)*cos(n*x), (x, 0, pi))
Respuesta (Indefinida) [src] 
                                                          //           2                      \                             
                                                          ||          x                       |                             
                                                          ||          --             for n = 0|                             
                                                          ||          2                       |                             
  /                            //   x      for n = 0\     ||                                  |       //   x      for n = 0\
 |                             ||                   |     ||/-cos(n*x)                        |       ||                   |
 | (3*x - 1)*cos(n*x) dx = C - |
$$\int \left(3 x - 1\right) \cos{\left(n x \right)}\, dx = C + 3 x \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right) - \begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(n x \right)}}{n} & \text{otherwise} \end{cases} - 3 \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: n = 0 \\\frac{\begin{cases} - \frac{\cos{\left(n x \right)}}{n} & \text{for}\: n \neq 0 \\0 & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases}\right)$$
Simplificar
Respuesta [src] 
/  3    sin(pi*n)   3*cos(pi*n)   3*pi*sin(pi*n)                                  
|- -- - --------- + ----------- + --------------  for And(n > -oo, n < oo, n != 0)
|   2       n             2             n                                         
|  n                     n                                                        
<                                                                                 
|                            2                                                    
|                        3*pi                                                     
|                  -pi + -----                               otherwise            
\                          2
$$\begin{cases} - \frac{\sin{\left(\pi n \right)}}{n} + \frac{3 \pi \sin{\left(\pi n \right)}}{n} + \frac{3 \cos{\left(\pi n \right)}}{n^{2}} - \frac{3}{n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\- \pi + \frac{3 \pi^{2}}{2} & \text{otherwise} \end{cases}$$ 
=
= 
/  3    sin(pi*n)   3*cos(pi*n)   3*pi*sin(pi*n)                                  
|- -- - --------- + ----------- + --------------  for And(n > -oo, n < oo, n != 0)
|   2       n             2             n                                         
|  n                     n                                                        
<                                                                                 
|                            2                                                    
|                        3*pi                                                     
|                  -pi + -----                               otherwise            
\                          2
$$\begin{cases} - \frac{\sin{\left(\pi n \right)}}{n} + \frac{3 \pi \sin{\left(\pi n \right)}}{n} + \frac{3 \cos{\left(\pi n \right)}}{n^{2}} - \frac{3}{n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\- \pi + \frac{3 \pi^{2}}{2} & \text{otherwise} \end{cases}$$ 
Piecewise((-3/n^2 - sin(pi*n)/n + 3*cos(pi*n)/n^2 + 3*pi*sin(pi*n)/n, (n > -oo)∧(n < oo)∧(Ne(n, 0))), (-pi + 3*pi^2/2, True))

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

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