Sr Examen

Otras calculadoras

Integral de (x-1)^2*cosnx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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

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