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Resolución de integrales/ (3x-2)/ 2/3*(3x-2)sin(p*n*x/3)

Integral de 2/3*(3x-2)sin(p*n*x/3) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  3                          
  /                          
 |                           
 |  2*(3*x - 2)    /p*n*x\   
 |  -----------*sin|-----| dx
 |       3         \  3  /   
 |                           
/                            
0
$$\int\limits_{0}^{3} \frac{2 \left(3 x - 2\right)}{3} \sin{\left(\frac{x n p}{3} \right)}\, dx$$ 
Integral((2*(3*x - 2)/3)*sin(((p*n)*x)/3), (x, 0, 3))
Respuesta (Indefinida) [src] 
                                                                                                                      //      0        for Or(n = 0, p = 0)\                                             
                                                                                                                      ||                                   |                                             
                                     //               0                  for Or(And(n = 0, p = 0), n = 0, p = 0)\     ||      /n*p*x\                      |                                             
                                     ||                                                                         |   4*|<-3*cos|-----|                      |                                             
  /                                  ||   //     /n*p*x\              \                                         |     ||      \  3  /                      |       //      0        for Or(n = 0, p = 0)\
 |                                   ||   ||3*sin|-----|              |                                         |     ||-------------       otherwise      |       ||                                   |
 | 2*(3*x - 2)    /p*n*x\            ||   ||     \  3  /      n*p     |                                         |     \\     n*p                           /       ||      /n*p*x\                      |
 | -----------*sin|-----| dx = C - 2*|<-3*|<------------  for --- != 0|                                         | - ---------------------------------------- + 2*x*|<-3*cos|-----|                      |
 |      3         \  3  /            ||   ||    n*p            3      |                                         |                      3                           ||      \  3  /                      |
 |                                   ||   ||                          |                                         |                                                  ||-------------       otherwise      |
/                                    ||   \\     x         otherwise  /                                         |                                                  \\     n*p                           /
                                     ||--------------------------------                 otherwise               |                                                                                        
                                     \\              n*p                                                        /
$$\int \frac{2 \left(3 x - 2\right)}{3} \sin{\left(\frac{x n p}{3} \right)}\, dx = C + 2 x \left(\begin{cases} 0 & \text{for}\: n = 0 \vee p = 0 \\- \frac{3 \cos{\left(\frac{n p x}{3} \right)}}{n p} & \text{otherwise} \end{cases}\right) - \frac{4 \left(\begin{cases} 0 & \text{for}\: n = 0 \vee p = 0 \\- \frac{3 \cos{\left(\frac{n p x}{3} \right)}}{n p} & \text{otherwise} \end{cases}\right)}{3} - 2 \left(\begin{cases} 0 & \text{for}\: \left(n = 0 \wedge p = 0\right) \vee n = 0 \vee p = 0 \\- \frac{3 \left(\begin{cases} \frac{3 \sin{\left(\frac{n p x}{3} \right)}}{n p} & \text{for}\: \frac{n p}{3} \neq 0 \\x & \text{otherwise} \end{cases}\right)}{n p} & \text{otherwise} \end{cases}\right)$$
Simplificar
Respuesta [src] 
/                0                  for Or(And(n = 0, p = 0), n = 0, p = 0)
|                                                                          
|   4    14*cos(n*p)   18*sin(n*p)                                         
<- --- - ----------- + -----------                 otherwise               
|  n*p       n*p           2  2                                            
|                         n *p                                             
\
$$\begin{cases} 0 & \text{for}\: \left(n = 0 \wedge p = 0\right) \vee n = 0 \vee p = 0 \\- \frac{14 \cos{\left(n p \right)}}{n p} - \frac{4}{n p} + \frac{18 \sin{\left(n p \right)}}{n^{2} p^{2}} & \text{otherwise} \end{cases}$$ 
=
= 
/                0                  for Or(And(n = 0, p = 0), n = 0, p = 0)
|                                                                          
|   4    14*cos(n*p)   18*sin(n*p)                                         
<- --- - ----------- + -----------                 otherwise               
|  n*p       n*p           2  2                                            
|                         n *p                                             
\
$$\begin{cases} 0 & \text{for}\: \left(n = 0 \wedge p = 0\right) \vee n = 0 \vee p = 0 \\- \frac{14 \cos{\left(n p \right)}}{n p} - \frac{4}{n p} + \frac{18 \sin{\left(n p \right)}}{n^{2} p^{2}} & \text{otherwise} \end{cases}$$ 
Piecewise((0, (n = 0)∨(p = 0)∨((n = 0)∧(p = 0))), (-4/(n*p) - 14*cos(n*p)/(n*p) + 18*sin(n*p)/(n^2*p^2), True))

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

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