Sr Examen

Integral de x^2cosnx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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

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