Sr Examen

Integral de x*sin(kx) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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

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