Sr Examen

Otras calculadoras

Resolución de integrales/ 0.5*x/ cos(x)/ cos(nx)/ 0.5*x*cos(x)*cos(nx)/pi

Integral de 0.5*x*cos(x)*cos(nx)/pi dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  2                     
  /                     
 |                      
 |  x                   
 |  -*cos(x)*cos(n*x)   
 |  2                   
 |  ----------------- dx
 |          pi          
 |                      
/                       
0
∫02x2cos⁡(x)cos⁡(nx)π dx\int\limits_{0}^{2} \frac{\frac{x}{2} \cos{\left(x \right)} \cos{\left(n x \right)}}{\pi}\, dx 
Integral((((x/2)*cos(x))*cos(n*x))/pi, (x, 0, 2))
Respuesta (Indefinida) [src] 
                              /                                                  2       2    2       2    2                                                                                            
                              |                                               cos (x)   x *cos (x)   x *sin (x)   x*cos(x)*sin(x)                                                                       
                              |                                               ------- + ---------- + ---------- + ---------------                                                  for Or(n = -1, n = 1)
                              |                                                  4          4            4               2                                                                              
                              |                                                                                                                                                                         
                              <                                       2                                            3                                            2                                       
  /                           |cos(x)*cos(n*x)   x*cos(n*x)*sin(x)   n *cos(x)*cos(n*x)   2*n*sin(x)*sin(n*x)   x*n *cos(x)*sin(n*x)   n*x*cos(x)*sin(n*x)   x*n *cos(n*x)*sin(x)                       
 |                            |--------------- + ----------------- + ------------------ + ------------------- + -------------------- - ------------------- - --------------------        otherwise      
 | x                          |      4      2           4      2            4      2              4      2              4      2               4      2              4      2                           
 | -*cos(x)*cos(n*x)          | 1 + n  - 2*n       1 + n  - 2*n        1 + n  - 2*n          1 + n  - 2*n          1 + n  - 2*n           1 + n  - 2*n          1 + n  - 2*n                            
 | 2                          \                                                                                                                                                                         
 | ----------------- dx = C + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------
 |         pi                                                                                                    2*pi                                                                                   
 |                                                                                                                                                                                                      
/
∫x2cos⁡(x)cos⁡(nx)π dx=C+{x2sin⁡2(x)4+x2cos⁡2(x)4+xsin⁡(x)cos⁡(x)2+cos⁡2(x)4for n=−1∨n=1n3xsin⁡(nx)cos⁡(x)n4−2n2+1−n2xsin⁡(x)cos⁡(nx)n4−2n2+1+n2cos⁡(x)cos⁡(nx)n4−2n2+1−nxsin⁡(nx)cos⁡(x)n4−2n2+1+2nsin⁡(x)sin⁡(nx)n4−2n2+1+xsin⁡(x)cos⁡(nx)n4−2n2+1+cos⁡(x)cos⁡(nx)n4−2n2+1otherwise2π\int \frac{\frac{x}{2} \cos{\left(x \right)} \cos{\left(n x \right)}}{\pi}\, dx = C + \frac{\begin{cases} \frac{x^{2} \sin^{2}{\left(x \right)}}{4} + \frac{x^{2} \cos^{2}{\left(x \right)}}{4} + \frac{x \sin{\left(x \right)} \cos{\left(x \right)}}{2} + \frac{\cos^{2}{\left(x \right)}}{4} & \text{for}\: n = -1 \vee n = 1 \\\frac{n^{3} x \sin{\left(n x \right)} \cos{\left(x \right)}}{n^{4} - 2 n^{2} + 1} - \frac{n^{2} x \sin{\left(x \right)} \cos{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} + \frac{n^{2} \cos{\left(x \right)} \cos{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} - \frac{n x \sin{\left(n x \right)} \cos{\left(x \right)}}{n^{4} - 2 n^{2} + 1} + \frac{2 n \sin{\left(x \right)} \sin{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} + \frac{x \sin{\left(x \right)} \cos{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} + \frac{\cos{\left(x \right)} \cos{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} & \text{otherwise} \end{cases}}{2 \pi}
Simplificar
Respuesta [src] 
/                                                                                      2                                                                                                                  
|                                                                          2      3*sin (2)                                                                                                               
|                                                                       cos (2) + --------- + cos(2)*sin(2)                                                                                               
|                                                                                     4                                                                                                                   
|                                                                       -----------------------------------                                                                          for Or(n = -1, n = 1)
|                                                                                       2*pi                                                                                                              
|                                                                                                                                                                                                         
<                                       2                                            2                                            3                                          2                            
|cos(2)*cos(2*n)   2*cos(2*n)*sin(2)   n *cos(2)*cos(2*n)   2*n*cos(2)*sin(2*n)   2*n *cos(2*n)*sin(2)   2*n*sin(2)*sin(2*n)   2*n *cos(2)*sin(2*n)         1               n                             
|--------------- + ----------------- + ------------------ - ------------------- - -------------------- + ------------------- + --------------------   ------------- + -------------                       
|      4      2           4      2            4      2              4      2              4      2               4      2              4      2            4      2        4      2                       
| 1 + n  - 2*n       1 + n  - 2*n        1 + n  - 2*n          1 + n  - 2*n          1 + n  - 2*n           1 + n  - 2*n          1 + n  - 2*n        1 + n  - 2*n    1 + n  - 2*n                        
|-------------------------------------------------------------------------------------------------------------------------------------------------- - -----------------------------        otherwise      
|                                                                       2*pi                                                                                       2*pi                                   
\
{sin⁡(2)cos⁡(2)+cos⁡2(2)+3sin⁡2(2)42πfor n=−1∨n=1−n2n4−2n2+1+1n4−2n2+12π+2n3sin⁡(2n)cos⁡(2)n4−2n2+1−2n2sin⁡(2)cos⁡(2n)n4−2n2+1+n2cos⁡(2)cos⁡(2n)n4−2n2+1−2nsin⁡(2n)cos⁡(2)n4−2n2+1+2nsin⁡(2)sin⁡(2n)n4−2n2+1+cos⁡(2)cos⁡(2n)n4−2n2+1+2sin⁡(2)cos⁡(2n)n4−2n2+12πotherwise\begin{cases} \frac{\sin{\left(2 \right)} \cos{\left(2 \right)} + \cos^{2}{\left(2 \right)} + \frac{3 \sin^{2}{\left(2 \right)}}{4}}{2 \pi} & \text{for}\: n = -1 \vee n = 1 \\- \frac{\frac{n^{2}}{n^{4} - 2 n^{2} + 1} + \frac{1}{n^{4} - 2 n^{2} + 1}}{2 \pi} + \frac{\frac{2 n^{3} \sin{\left(2 n \right)} \cos{\left(2 \right)}}{n^{4} - 2 n^{2} + 1} - \frac{2 n^{2} \sin{\left(2 \right)} \cos{\left(2 n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{n^{2} \cos{\left(2 \right)} \cos{\left(2 n \right)}}{n^{4} - 2 n^{2} + 1} - \frac{2 n \sin{\left(2 n \right)} \cos{\left(2 \right)}}{n^{4} - 2 n^{2} + 1} + \frac{2 n \sin{\left(2 \right)} \sin{\left(2 n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{\cos{\left(2 \right)} \cos{\left(2 n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{2 \sin{\left(2 \right)} \cos{\left(2 n \right)}}{n^{4} - 2 n^{2} + 1}}{2 \pi} & \text{otherwise} \end{cases} 
=
= 
/                                                                                      2                                                                                                                  
|                                                                          2      3*sin (2)                                                                                                               
|                                                                       cos (2) + --------- + cos(2)*sin(2)                                                                                               
|                                                                                     4                                                                                                                   
|                                                                       -----------------------------------                                                                          for Or(n = -1, n = 1)
|                                                                                       2*pi                                                                                                              
|                                                                                                                                                                                                         
<                                       2                                            2                                            3                                          2                            
|cos(2)*cos(2*n)   2*cos(2*n)*sin(2)   n *cos(2)*cos(2*n)   2*n*cos(2)*sin(2*n)   2*n *cos(2*n)*sin(2)   2*n*sin(2)*sin(2*n)   2*n *cos(2)*sin(2*n)         1               n                             
|--------------- + ----------------- + ------------------ - ------------------- - -------------------- + ------------------- + --------------------   ------------- + -------------                       
|      4      2           4      2            4      2              4      2              4      2               4      2              4      2            4      2        4      2                       
| 1 + n  - 2*n       1 + n  - 2*n        1 + n  - 2*n          1 + n  - 2*n          1 + n  - 2*n           1 + n  - 2*n          1 + n  - 2*n        1 + n  - 2*n    1 + n  - 2*n                        
|-------------------------------------------------------------------------------------------------------------------------------------------------- - -----------------------------        otherwise      
|                                                                       2*pi                                                                                       2*pi                                   
\
{sin⁡(2)cos⁡(2)+cos⁡2(2)+3sin⁡2(2)42πfor n=−1∨n=1−n2n4−2n2+1+1n4−2n2+12π+2n3sin⁡(2n)cos⁡(2)n4−2n2+1−2n2sin⁡(2)cos⁡(2n)n4−2n2+1+n2cos⁡(2)cos⁡(2n)n4−2n2+1−2nsin⁡(2n)cos⁡(2)n4−2n2+1+2nsin⁡(2)sin⁡(2n)n4−2n2+1+cos⁡(2)cos⁡(2n)n4−2n2+1+2sin⁡(2)cos⁡(2n)n4−2n2+12πotherwise\begin{cases} \frac{\sin{\left(2 \right)} \cos{\left(2 \right)} + \cos^{2}{\left(2 \right)} + \frac{3 \sin^{2}{\left(2 \right)}}{4}}{2 \pi} & \text{for}\: n = -1 \vee n = 1 \\- \frac{\frac{n^{2}}{n^{4} - 2 n^{2} + 1} + \frac{1}{n^{4} - 2 n^{2} + 1}}{2 \pi} + \frac{\frac{2 n^{3} \sin{\left(2 n \right)} \cos{\left(2 \right)}}{n^{4} - 2 n^{2} + 1} - \frac{2 n^{2} \sin{\left(2 \right)} \cos{\left(2 n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{n^{2} \cos{\left(2 \right)} \cos{\left(2 n \right)}}{n^{4} - 2 n^{2} + 1} - \frac{2 n \sin{\left(2 n \right)} \cos{\left(2 \right)}}{n^{4} - 2 n^{2} + 1} + \frac{2 n \sin{\left(2 \right)} \sin{\left(2 n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{\cos{\left(2 \right)} \cos{\left(2 n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{2 \sin{\left(2 \right)} \cos{\left(2 n \right)}}{n^{4} - 2 n^{2} + 1}}{2 \pi} & \text{otherwise} \end{cases} 
Piecewise(((cos(2)^2 + 3*sin(2)^2/4 + cos(2)*sin(2))/(2*pi), (n = -1)∨(n = 1)), ((cos(2)*cos(2*n)/(1 + n^4 - 2*n^2) + 2*cos(2*n)*sin(2)/(1 + n^4 - 2*n^2) + n^2*cos(2)*cos(2*n)/(1 + n^4 - 2*n^2) - 2*n*cos(2)*sin(2*n)/(1 + n^4 - 2*n^2) - 2*n^2*cos(2*n)*sin(2)/(1 + n^4 - 2*n^2) + 2*n*sin(2)*sin(2*n)/(1 + n^4 - 2*n^2) + 2*n^3*cos(2)*sin(2*n)/(1 + n^4 - 2*n^2))/(2*pi) - (1/(1 + n^4 - 2*n^2) + n^2/(1 + n^4 - 2*n^2))/(2*pi), True))

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

    © Sr Examen – Калькулятор