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Resolución de integrales/ 5*pi*x*cos((5*pi)/8)*sin((m*x)/7)

Integral de 5*pi*x*cos((5*pi)/8)*sin((m*x)/7) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  7                             
  /                             
 |                              
 |            /5*pi\    /m*x\   
 |  5*pi*x*cos|----|*sin|---| dx
 |            \ 8  /    \ 7 /   
 |                              
/                               
0
∫075πxcos⁡(5π8)sin⁡(mx7) dx\int\limits_{0}^{7} 5 \pi x \cos{\left(\frac{5 \pi}{8} \right)} \sin{\left(\frac{m x}{7} \right)}\, dx 
Integral((((5*pi)*x)*cos((5*pi)/8))*sin((m*x)/7), (x, 0, 7))
Respuesta (Indefinida) [src] 
                                                          /  //             0                for m = 0\                              \
                                                          |  ||                                       |                              |
                                                          |  ||   //     /m*x\            \           |     //     0       for m = 0\|
                                              ___________ |  ||   ||7*sin|---|            |           |     ||                      ||
                                             /       ___  |  ||   ||     \ 7 /      m     |           |     ||      /m*x\           ||
                                      5*pi*\/  2 - \/ 2  *|- |<-7*|<----------  for - != 0|           | + x*|<-7*cos|---|           ||
                                                          |  ||   ||    m           7     |           |     ||      \ 7 /           ||
                                                          |  ||   ||                      |           |     ||-----------  otherwise||
  /                                                       |  ||   \\    x       otherwise /           |     \\     m                /|
 |                                                        |  ||----------------------------  otherwise|                              |
 |           /5*pi\    /m*x\                              \  \\             m                         /                              /
 | 5*pi*x*cos|----|*sin|---| dx = C - ------------------------------------------------------------------------------------------------
 |           \ 8  /    \ 7 /                                                         2                                                
 |                                                                                                                                    
/
∫5πxcos⁡(5π8)sin⁡(mx7) dx=C−5π2−2(x({0for m=0−7cos⁡(mx7)motherwise)−{0for m=0−7({7sin⁡(mx7)mfor m7≠0xotherwise)motherwise)2\int 5 \pi x \cos{\left(\frac{5 \pi}{8} \right)} \sin{\left(\frac{m x}{7} \right)}\, dx = C - \frac{5 \pi \sqrt{2 - \sqrt{2}} \left(x \left(\begin{cases} 0 & \text{for}\: m = 0 \\- \frac{7 \cos{\left(\frac{m x}{7} \right)}}{m} & \text{otherwise} \end{cases}\right) - \begin{cases} 0 & \text{for}\: m = 0 \\- \frac{7 \left(\begin{cases} \frac{7 \sin{\left(\frac{m x}{7} \right)}}{m} & \text{for}\: \frac{m}{7} \neq 0 \\x & \text{otherwise} \end{cases}\right)}{m} & \text{otherwise} \end{cases}\right)}{2}
Simplificar
Respuesta [src] 
/           ___________                                                            
|          /       ___                                                             
|         /  1   \/ 2   /  49*cos(m)   49*sin(m)\                                  
|-5*pi*  /   - - ----- *|- --------- + ---------|  for And(m > -oo, m < oo, m != 0)
<      \/    2     4    |      m            2   |                                  
|                       \                  m    /                                  
|                                                                                  
|                       0                                     otherwise            
\
{−5π12−24(−49cos⁡(m)m+49sin⁡(m)m2)for m>−∞∧m<∞∧m≠00otherwise\begin{cases} - 5 \pi \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} \left(- \frac{49 \cos{\left(m \right)}}{m} + \frac{49 \sin{\left(m \right)}}{m^{2}}\right) & \text{for}\: m > -\infty \wedge m < \infty \wedge m \neq 0 \\0 & \text{otherwise} \end{cases} 
=
= 
/           ___________                                                            
|          /       ___                                                             
|         /  1   \/ 2   /  49*cos(m)   49*sin(m)\                                  
|-5*pi*  /   - - ----- *|- --------- + ---------|  for And(m > -oo, m < oo, m != 0)
<      \/    2     4    |      m            2   |                                  
|                       \                  m    /                                  
|                                                                                  
|                       0                                     otherwise            
\
{−5π12−24(−49cos⁡(m)m+49sin⁡(m)m2)for m>−∞∧m<∞∧m≠00otherwise\begin{cases} - 5 \pi \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} \left(- \frac{49 \cos{\left(m \right)}}{m} + \frac{49 \sin{\left(m \right)}}{m^{2}}\right) & \text{for}\: m > -\infty \wedge m < \infty \wedge m \neq 0 \\0 & \text{otherwise} \end{cases} 
Piecewise((-5*pi*sqrt(1/2 - sqrt(2)/4)*(-49*cos(m)/m + 49*sin(m)/m^2), (m > -oo)∧(m < oo)∧(Ne(m, 0))), (0, True))

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

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