Sr Examen

Integral de xsinnx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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

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