Sr Examen

Integral de sin(x)*cos(nx) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
  0                   
  /                   
 |                    
 |  sin(x)*cos(n*x) dx
 |                    
/                     
-pi                   
$$\int\limits_{- \pi}^{0} \sin{\left(x \right)} \cos{\left(n x \right)}\, dx$$
Integral(sin(x)*cos(n*x), (x, -pi, 0))
Respuesta (Indefinida) [src]
                            //                 2                                        \
                            ||             -cos (x)                                     |
  /                         ||             ---------               for Or(n = -1, n = 1)|
 |                          ||                 2                                        |
 | sin(x)*cos(n*x) dx = C + |<                                                          |
 |                          ||cos(x)*cos(n*x)   n*sin(x)*sin(n*x)                       |
/                           ||--------------- + -----------------        otherwise      |
                            ||          2                  2                            |
                            \\    -1 + n             -1 + n                             /
$$\int \sin{\left(x \right)} \cos{\left(n x \right)}\, dx = C + \begin{cases} - \frac{\cos^{2}{\left(x \right)}}{2} & \text{for}\: n = -1 \vee n = 1 \\\frac{n \sin{\left(x \right)} \sin{\left(n x \right)}}{n^{2} - 1} + \frac{\cos{\left(x \right)} \cos{\left(n x \right)}}{n^{2} - 1} & \text{otherwise} \end{cases}$$
Respuesta [src]
/         0           for Or(n = -1, n = 1)
|                                          
|   1      cos(pi*n)                       
<------- + ---------        otherwise      
|      2          2                        
|-1 + n     -1 + n                         
\                                          
$$\begin{cases} 0 & \text{for}\: n = -1 \vee n = 1 \\\frac{\cos{\left(\pi n \right)}}{n^{2} - 1} + \frac{1}{n^{2} - 1} & \text{otherwise} \end{cases}$$
=
=
/         0           for Or(n = -1, n = 1)
|                                          
|   1      cos(pi*n)                       
<------- + ---------        otherwise      
|      2          2                        
|-1 + n     -1 + n                         
\                                          
$$\begin{cases} 0 & \text{for}\: n = -1 \vee n = 1 \\\frac{\cos{\left(\pi n \right)}}{n^{2} - 1} + \frac{1}{n^{2} - 1} & \text{otherwise} \end{cases}$$
Piecewise((0, (n = -1)∨(n = 1)), (1/(-1 + n^2) + cos(pi*n)/(-1 + n^2), True))

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