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Resolución de integrales/ cosnx/ sinxcos/ sinxcosnx

Integral de sinxcosnx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                   
  /                   
 |                    
 |  sin(x)*cos(n*x) dx
 |                    
/                     
0
∫01sin⁡(x)cos⁡(nx) dx\int\limits_{0}^{1} \sin{\left(x \right)} \cos{\left(n x \right)}\, dx 
Integral(sin(x)*cos(n*x), (x, 0, 1))
Respuesta (Indefinida) [src] 
                            //                 2                                        \
                            ||              sin (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                             /
∫sin⁡(x)cos⁡(nx) dx=C+{sin⁡2(x)2for n=−1∨n=1nsin⁡(x)sin⁡(nx)n2−1+cos⁡(x)cos⁡(nx)n2−1otherwise\int \sin{\left(x \right)} \cos{\left(n x \right)}\, dx = C + \begin{cases} \frac{\sin^{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}
Simplificar
Respuesta [src] 
/                     2                                            
|                  sin (1)                                         
|                  -------                    for Or(n = -1, n = 1)
|                     2                                            
<                                                                  
|     1      cos(1)*cos(n)   n*sin(1)*sin(n)                       
|- ------- + ------------- + ---------------        otherwise      
|        2            2                2                           
\  -1 + n       -1 + n           -1 + n
{sin⁡2(1)2for n=−1∨n=1nsin⁡(1)sin⁡(n)n2−1+cos⁡(1)cos⁡(n)n2−1−1n2−1otherwise\begin{cases} \frac{\sin^{2}{\left(1 \right)}}{2} & \text{for}\: n = -1 \vee n = 1 \\\frac{n \sin{\left(1 \right)} \sin{\left(n \right)}}{n^{2} - 1} + \frac{\cos{\left(1 \right)} \cos{\left(n \right)}}{n^{2} - 1} - \frac{1}{n^{2} - 1} & \text{otherwise} \end{cases} 
=
= 
/                     2                                            
|                  sin (1)                                         
|                  -------                    for Or(n = -1, n = 1)
|                     2                                            
<                                                                  
|     1      cos(1)*cos(n)   n*sin(1)*sin(n)                       
|- ------- + ------------- + ---------------        otherwise      
|        2            2                2                           
\  -1 + n       -1 + n           -1 + n
{sin⁡2(1)2for n=−1∨n=1nsin⁡(1)sin⁡(n)n2−1+cos⁡(1)cos⁡(n)n2−1−1n2−1otherwise\begin{cases} \frac{\sin^{2}{\left(1 \right)}}{2} & \text{for}\: n = -1 \vee n = 1 \\\frac{n \sin{\left(1 \right)} \sin{\left(n \right)}}{n^{2} - 1} + \frac{\cos{\left(1 \right)} \cos{\left(n \right)}}{n^{2} - 1} - \frac{1}{n^{2} - 1} & \text{otherwise} \end{cases} 
Piecewise((sin(1)^2/2, (n = -1)∨(n = 1)), (-1/(-1 + n^2) + cos(1)*cos(n)/(-1 + n^2) + n*sin(1)*sin(n)/(-1 + n^2), True))

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

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