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Resolución de integrales/ cosnx/ cosnxcosmx

Integral de cosnxcosmx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 pi                     
  /                     
 |                      
 |  cos(n*x)*cos(m*x) dx
 |                      
/                       
-pi
∫−ππcos⁡(mx)cos⁡(nx) dx\int\limits_{- \pi}^{\pi} \cos{\left(m x \right)} \cos{\left(n x \right)}\, dx 
Integral(cos(n*x)*cos(m*x), (x, -pi, pi))
Respuesta (Indefinida) [src] 
                              //                      x                        for And(m = 0, n = 0)\
                              ||                                                                    |
                              ||     2             2                                                |
  /                           ||x*cos (n*x)   x*sin (n*x)   cos(n*x)*sin(n*x)                       |
 |                            ||----------- + ----------- + -----------------  for Or(m = -n, m = n)|
 | cos(n*x)*cos(m*x) dx = C + |<     2             2               2*n                              |
 |                            ||                                                                    |
/                             ||  m*cos(n*x)*sin(m*x)   n*cos(m*x)*sin(n*x)                         |
                              ||  ------------------- - -------------------          otherwise      |
                              ||         2    2                2    2                               |
                              \\        m  - n                m  - n                                /
∫cos⁡(mx)cos⁡(nx) dx=C+{xfor m=0∧n=0xsin⁡2(nx)2+xcos⁡2(nx)2+sin⁡(nx)cos⁡(nx)2nfor m=−n∨m=nmsin⁡(mx)cos⁡(nx)m2−n2−nsin⁡(nx)cos⁡(mx)m2−n2otherwise\int \cos{\left(m x \right)} \cos{\left(n x \right)}\, dx = C + \begin{cases} x & \text{for}\: m = 0 \wedge n = 0 \\\frac{x \sin^{2}{\left(n x \right)}}{2} + \frac{x \cos^{2}{\left(n x \right)}}{2} + \frac{\sin{\left(n x \right)} \cos{\left(n x \right)}}{2 n} & \text{for}\: m = - n \vee m = n \\\frac{m \sin{\left(m x \right)} \cos{\left(n x \right)}}{m^{2} - n^{2}} - \frac{n \sin{\left(n x \right)} \cos{\left(m x \right)}}{m^{2} - n^{2}} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/                       2*pi                                                        for Or(And(m = 0, n = 0), And(m = 0, m = n, n = 0), And(m = 0, m = -n, n = 0), And(m = 0, m = -n, m = n, n = 0))                               
|                                                                                                                                                                                                                                  
|      2               2         cos(pi*n)*sin(pi*n)                                                                                                                                                                               
|pi*cos (pi*n) + pi*sin (pi*n) + -------------------  for Or(And(m = 0, m = -n), And(m = 0, m = n), And(m = -n, m = n), And(m = n, n = 0), And(m = -n, n = 0), And(m = 0, m = -n, m = n), And(m = -n, m = n, n = 0), m = -n, m = n)
|                                         n                                                                                                                                                                                        
<                                                                                                                                                                                                                                  
|  2*n*cos(pi*m)*sin(pi*n)   2*m*cos(pi*n)*sin(pi*m)                                                                                                                                                                               
|- ----------------------- + -----------------------                                                                                    otherwise                                                                                  
|           2    2                    2    2                                                                                                                                                                                       
|          m  - n                    m  - n                                                                                                                                                                                        
\
{2πfor (m=0∧n=0)∨(m=0∧m=n∧n=0)∨(m=0∧m=−n∧n=0)∨(m=0∧m=−n∧m=n∧n=0)πsin⁡2(πn)+πcos⁡2(πn)+sin⁡(πn)cos⁡(πn)nfor (m=0∧m=−n)∨(m=0∧m=n)∨(m=−n∧m=n)∨(m=n∧n=0)∨(m=−n∧n=0)∨(m=0∧m=−n∧m=n)∨(m=−n∧m=n∧n=0)∨m=−n∨m=n2msin⁡(πm)cos⁡(πn)m2−n2−2nsin⁡(πn)cos⁡(πm)m2−n2otherwise\begin{cases} 2 \pi & \text{for}\: \left(m = 0 \wedge n = 0\right) \vee \left(m = 0 \wedge m = n \wedge n = 0\right) \vee \left(m = 0 \wedge m = - n \wedge n = 0\right) \vee \left(m = 0 \wedge m = - n \wedge m = n \wedge n = 0\right) \\\pi \sin^{2}{\left(\pi n \right)} + \pi \cos^{2}{\left(\pi n \right)} + \frac{\sin{\left(\pi n \right)} \cos{\left(\pi n \right)}}{n} & \text{for}\: \left(m = 0 \wedge m = - n\right) \vee \left(m = 0 \wedge m = n\right) \vee \left(m = - n \wedge m = n\right) \vee \left(m = n \wedge n = 0\right) \vee \left(m = - n \wedge n = 0\right) \vee \left(m = 0 \wedge m = - n \wedge m = n\right) \vee \left(m = - n \wedge m = n \wedge n = 0\right) \vee m = - n \vee m = n \\\frac{2 m \sin{\left(\pi m \right)} \cos{\left(\pi n \right)}}{m^{2} - n^{2}} - \frac{2 n \sin{\left(\pi n \right)} \cos{\left(\pi m \right)}}{m^{2} - n^{2}} & \text{otherwise} \end{cases} 
=
= 
/                       2*pi                                                        for Or(And(m = 0, n = 0), And(m = 0, m = n, n = 0), And(m = 0, m = -n, n = 0), And(m = 0, m = -n, m = n, n = 0))                               
|                                                                                                                                                                                                                                  
|      2               2         cos(pi*n)*sin(pi*n)                                                                                                                                                                               
|pi*cos (pi*n) + pi*sin (pi*n) + -------------------  for Or(And(m = 0, m = -n), And(m = 0, m = n), And(m = -n, m = n), And(m = n, n = 0), And(m = -n, n = 0), And(m = 0, m = -n, m = n), And(m = -n, m = n, n = 0), m = -n, m = n)
|                                         n                                                                                                                                                                                        
<                                                                                                                                                                                                                                  
|  2*n*cos(pi*m)*sin(pi*n)   2*m*cos(pi*n)*sin(pi*m)                                                                                                                                                                               
|- ----------------------- + -----------------------                                                                                    otherwise                                                                                  
|           2    2                    2    2                                                                                                                                                                                       
|          m  - n                    m  - n                                                                                                                                                                                        
\
{2πfor (m=0∧n=0)∨(m=0∧m=n∧n=0)∨(m=0∧m=−n∧n=0)∨(m=0∧m=−n∧m=n∧n=0)πsin⁡2(πn)+πcos⁡2(πn)+sin⁡(πn)cos⁡(πn)nfor (m=0∧m=−n)∨(m=0∧m=n)∨(m=−n∧m=n)∨(m=n∧n=0)∨(m=−n∧n=0)∨(m=0∧m=−n∧m=n)∨(m=−n∧m=n∧n=0)∨m=−n∨m=n2msin⁡(πm)cos⁡(πn)m2−n2−2nsin⁡(πn)cos⁡(πm)m2−n2otherwise\begin{cases} 2 \pi & \text{for}\: \left(m = 0 \wedge n = 0\right) \vee \left(m = 0 \wedge m = n \wedge n = 0\right) \vee \left(m = 0 \wedge m = - n \wedge n = 0\right) \vee \left(m = 0 \wedge m = - n \wedge m = n \wedge n = 0\right) \\\pi \sin^{2}{\left(\pi n \right)} + \pi \cos^{2}{\left(\pi n \right)} + \frac{\sin{\left(\pi n \right)} \cos{\left(\pi n \right)}}{n} & \text{for}\: \left(m = 0 \wedge m = - n\right) \vee \left(m = 0 \wedge m = n\right) \vee \left(m = - n \wedge m = n\right) \vee \left(m = n \wedge n = 0\right) \vee \left(m = - n \wedge n = 0\right) \vee \left(m = 0 \wedge m = - n \wedge m = n\right) \vee \left(m = - n \wedge m = n \wedge n = 0\right) \vee m = - n \vee m = n \\\frac{2 m \sin{\left(\pi m \right)} \cos{\left(\pi n \right)}}{m^{2} - n^{2}} - \frac{2 n \sin{\left(\pi n \right)} \cos{\left(\pi m \right)}}{m^{2} - n^{2}} & \text{otherwise} \end{cases} 
Piecewise((2*pi, ((m = 0)∧(n = 0))∨((m = 0)∧(m = n)∧(n = 0))∨((m = 0)∧(n = 0)∧(m = -n))∨((m = 0)∧(m = n)∧(n = 0)∧(m = -n))), (pi*cos(pi*n)^2 + pi*sin(pi*n)^2 + cos(pi*n)*sin(pi*n)/n, (m = n)∨(m = -n)∨((m = 0)∧(m = n))∨((m = n)∧(n = 0))∨((m = 0)∧(m = -n))∨((m = n)∧(m = -n))∨((n = 0)∧(m = -n))∨((m = 0)∧(m = n)∧(m = -n))∨((m = n)∧(n = 0)∧(m = -n))), (-2*n*cos(pi*m)*sin(pi*n)/(m^2 - n^2) + 2*m*cos(pi*n)*sin(pi*m)/(m^2 - n^2), True))

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

    © Sr Examen – Калькулятор