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Resolución de integrales/ cos(x)/ sin(n*x)/ ((cos(x))^2)*sin(n*x)

Integral de ((cos(x))^2)*sin(n*x) dx

Límites de integración:

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v

Gráfico:

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Definida a trozos:

Solución

Ha introducido [src] 
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∫01sin⁡(nx)cos⁡2(x) dx\int\limits_{0}^{1} \sin{\left(n x \right)} \cos^{2}{\left(x \right)}\, dx 
Integral(cos(x)^2*sin(n*x), (x, 0, 1))
Respuesta (Indefinida) [src] 
                             //    2                                               2                    2                                                    \
                             || sin (x)*cos(2*x)   3*cos(x)*sin(x)*sin(2*x)   x*cos (x)*sin(2*x)   x*sin (x)*sin(2*x)   x*cos(x)*cos(2*x)*sin(x)             |
                             || ---------------- - ------------------------ - ------------------ + ------------------ + ------------------------   for n = -2|
                             ||        2                      4                       4                    4                       2                         |
                             ||                                                                                                                              |
  /                          ||                                                        0                                                           for n = 0 |
 |                           ||                                                                                                                              |
 |    2                      ||     2                    2                    2                                                                              |
 | cos (x)*sin(n*x) dx = C + |<  sin (x)*cos(2*x)   x*sin (x)*sin(2*x)   x*cos (x)*sin(2*x)   3*cos(x)*sin(x)*sin(2*x)   x*cos(x)*cos(2*x)*sin(x)            |
 |                           ||- ---------------- - ------------------ + ------------------ + ------------------------ - ------------------------  for n = 2 |
/                            ||         2                   4                    4                       4                          2                        |
                             ||                                                                                                                              |
                             ||                 2                    2                2    2                                                                 |
                             ||            2*cos (x)*cos(n*x)   2*sin (x)*cos(n*x)   n *cos (x)*cos(n*x)   2*n*cos(x)*sin(x)*sin(n*x)                        |
                             ||            ------------------ + ------------------ - ------------------- - --------------------------              otherwise |
                             ||                  3                    3                     3                        3                                       |
                             \\                 n  - 4*n             n  - 4*n              n  - 4*n                 n  - 4*n                                 /
∫sin⁡(nx)cos⁡2(x) dx=C+{xsin⁡2(x)sin⁡(2x)4+xsin⁡(x)cos⁡(x)cos⁡(2x)2−xsin⁡(2x)cos⁡2(x)4+sin⁡2(x)cos⁡(2x)2−3sin⁡(x)sin⁡(2x)cos⁡(x)4for n=−20for n=0−xsin⁡2(x)sin⁡(2x)4−xsin⁡(x)cos⁡(x)cos⁡(2x)2+xsin⁡(2x)cos⁡2(x)4−sin⁡2(x)cos⁡(2x)2+3sin⁡(x)sin⁡(2x)cos⁡(x)4for n=2−n2cos⁡2(x)cos⁡(nx)n3−4n−2nsin⁡(x)sin⁡(nx)cos⁡(x)n3−4n+2sin⁡2(x)cos⁡(nx)n3−4n+2cos⁡2(x)cos⁡(nx)n3−4notherwise\int \sin{\left(n x \right)} \cos^{2}{\left(x \right)}\, dx = C + \begin{cases} \frac{x \sin^{2}{\left(x \right)} \sin{\left(2 x \right)}}{4} + \frac{x \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(2 x \right)}}{2} - \frac{x \sin{\left(2 x \right)} \cos^{2}{\left(x \right)}}{4} + \frac{\sin^{2}{\left(x \right)} \cos{\left(2 x \right)}}{2} - \frac{3 \sin{\left(x \right)} \sin{\left(2 x \right)} \cos{\left(x \right)}}{4} & \text{for}\: n = -2 \\0 & \text{for}\: n = 0 \\- \frac{x \sin^{2}{\left(x \right)} \sin{\left(2 x \right)}}{4} - \frac{x \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(2 x \right)}}{2} + \frac{x \sin{\left(2 x \right)} \cos^{2}{\left(x \right)}}{4} - \frac{\sin^{2}{\left(x \right)} \cos{\left(2 x \right)}}{2} + \frac{3 \sin{\left(x \right)} \sin{\left(2 x \right)} \cos{\left(x \right)}}{4} & \text{for}\: n = 2 \\- \frac{n^{2} \cos^{2}{\left(x \right)} \cos{\left(n x \right)}}{n^{3} - 4 n} - \frac{2 n \sin{\left(x \right)} \sin{\left(n x \right)} \cos{\left(x \right)}}{n^{3} - 4 n} + \frac{2 \sin^{2}{\left(x \right)} \cos{\left(n x \right)}}{n^{3} - 4 n} + \frac{2 \cos^{2}{\left(x \right)} \cos{\left(n x \right)}}{n^{3} - 4 n} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/            2                2                2                                                                       
|     1   cos (1)*cos(2)   cos (1)*sin(2)   sin (1)*sin(2)   cos(1)*cos(2)*sin(1)   cos(1)*sin(1)*sin(2)               
|   - - + -------------- - -------------- + -------------- + -------------------- + --------------------     for n = -2
|     2         2                4                4                   2                      4                         
|                                                                                                                      
|                                                    0                                                       for n = 0 
|                                                                                                                      
|           2                2                2                                                                        
<    1   cos (1)*cos(2)   sin (1)*sin(2)   cos (1)*sin(2)   cos(1)*cos(2)*sin(1)   cos(1)*sin(1)*sin(2)                
|    - - -------------- - -------------- + -------------- - -------------------- - --------------------      for n = 2 
|    2         2                4                4                   2                      4                          
|                                                                                                                      
|                 2           2                  2              2    2                                                 
|     2          n       2*cos (1)*cos(n)   2*sin (1)*cos(n)   n *cos (1)*cos(n)   2*n*cos(1)*sin(1)*sin(n)            
|- -------- + -------- + ---------------- + ---------------- - ----------------- - ------------------------  otherwise 
|   3          3              3                  3                   3                      3                          
\  n  - 4*n   n  - 4*n       n  - 4*n           n  - 4*n            n  - 4*n               n  - 4*n
{−12+sin⁡(1)cos⁡(1)cos⁡(2)2−sin⁡(2)cos⁡2(1)4+cos⁡2(1)cos⁡(2)2+sin⁡(1)sin⁡(2)cos⁡(1)4+sin⁡2(1)sin⁡(2)4for n=−20for n=0−sin⁡2(1)sin⁡(2)4−sin⁡(1)sin⁡(2)cos⁡(1)4−cos⁡2(1)cos⁡(2)2+sin⁡(2)cos⁡2(1)4−sin⁡(1)cos⁡(1)cos⁡(2)2+12for n=2−n2cos⁡2(1)cos⁡(n)n3−4n+n2n3−4n−2nsin⁡(1)sin⁡(n)cos⁡(1)n3−4n+2cos⁡2(1)cos⁡(n)n3−4n+2sin⁡2(1)cos⁡(n)n3−4n−2n3−4notherwise\begin{cases} - \frac{1}{2} + \frac{\sin{\left(1 \right)} \cos{\left(1 \right)} \cos{\left(2 \right)}}{2} - \frac{\sin{\left(2 \right)} \cos^{2}{\left(1 \right)}}{4} + \frac{\cos^{2}{\left(1 \right)} \cos{\left(2 \right)}}{2} + \frac{\sin{\left(1 \right)} \sin{\left(2 \right)} \cos{\left(1 \right)}}{4} + \frac{\sin^{2}{\left(1 \right)} \sin{\left(2 \right)}}{4} & \text{for}\: n = -2 \\0 & \text{for}\: n = 0 \\- \frac{\sin^{2}{\left(1 \right)} \sin{\left(2 \right)}}{4} - \frac{\sin{\left(1 \right)} \sin{\left(2 \right)} \cos{\left(1 \right)}}{4} - \frac{\cos^{2}{\left(1 \right)} \cos{\left(2 \right)}}{2} + \frac{\sin{\left(2 \right)} \cos^{2}{\left(1 \right)}}{4} - \frac{\sin{\left(1 \right)} \cos{\left(1 \right)} \cos{\left(2 \right)}}{2} + \frac{1}{2} & \text{for}\: n = 2 \\- \frac{n^{2} \cos^{2}{\left(1 \right)} \cos{\left(n \right)}}{n^{3} - 4 n} + \frac{n^{2}}{n^{3} - 4 n} - \frac{2 n \sin{\left(1 \right)} \sin{\left(n \right)} \cos{\left(1 \right)}}{n^{3} - 4 n} + \frac{2 \cos^{2}{\left(1 \right)} \cos{\left(n \right)}}{n^{3} - 4 n} + \frac{2 \sin^{2}{\left(1 \right)} \cos{\left(n \right)}}{n^{3} - 4 n} - \frac{2}{n^{3} - 4 n} & \text{otherwise} \end{cases} 
=
= 
/            2                2                2                                                                       
|     1   cos (1)*cos(2)   cos (1)*sin(2)   sin (1)*sin(2)   cos(1)*cos(2)*sin(1)   cos(1)*sin(1)*sin(2)               
|   - - + -------------- - -------------- + -------------- + -------------------- + --------------------     for n = -2
|     2         2                4                4                   2                      4                         
|                                                                                                                      
|                                                    0                                                       for n = 0 
|                                                                                                                      
|           2                2                2                                                                        
<    1   cos (1)*cos(2)   sin (1)*sin(2)   cos (1)*sin(2)   cos(1)*cos(2)*sin(1)   cos(1)*sin(1)*sin(2)                
|    - - -------------- - -------------- + -------------- - -------------------- - --------------------      for n = 2 
|    2         2                4                4                   2                      4                          
|                                                                                                                      
|                 2           2                  2              2    2                                                 
|     2          n       2*cos (1)*cos(n)   2*sin (1)*cos(n)   n *cos (1)*cos(n)   2*n*cos(1)*sin(1)*sin(n)            
|- -------- + -------- + ---------------- + ---------------- - ----------------- - ------------------------  otherwise 
|   3          3              3                  3                   3                      3                          
\  n  - 4*n   n  - 4*n       n  - 4*n           n  - 4*n            n  - 4*n               n  - 4*n
{−12+sin⁡(1)cos⁡(1)cos⁡(2)2−sin⁡(2)cos⁡2(1)4+cos⁡2(1)cos⁡(2)2+sin⁡(1)sin⁡(2)cos⁡(1)4+sin⁡2(1)sin⁡(2)4for n=−20for n=0−sin⁡2(1)sin⁡(2)4−sin⁡(1)sin⁡(2)cos⁡(1)4−cos⁡2(1)cos⁡(2)2+sin⁡(2)cos⁡2(1)4−sin⁡(1)cos⁡(1)cos⁡(2)2+12for n=2−n2cos⁡2(1)cos⁡(n)n3−4n+n2n3−4n−2nsin⁡(1)sin⁡(n)cos⁡(1)n3−4n+2cos⁡2(1)cos⁡(n)n3−4n+2sin⁡2(1)cos⁡(n)n3−4n−2n3−4notherwise\begin{cases} - \frac{1}{2} + \frac{\sin{\left(1 \right)} \cos{\left(1 \right)} \cos{\left(2 \right)}}{2} - \frac{\sin{\left(2 \right)} \cos^{2}{\left(1 \right)}}{4} + \frac{\cos^{2}{\left(1 \right)} \cos{\left(2 \right)}}{2} + \frac{\sin{\left(1 \right)} \sin{\left(2 \right)} \cos{\left(1 \right)}}{4} + \frac{\sin^{2}{\left(1 \right)} \sin{\left(2 \right)}}{4} & \text{for}\: n = -2 \\0 & \text{for}\: n = 0 \\- \frac{\sin^{2}{\left(1 \right)} \sin{\left(2 \right)}}{4} - \frac{\sin{\left(1 \right)} \sin{\left(2 \right)} \cos{\left(1 \right)}}{4} - \frac{\cos^{2}{\left(1 \right)} \cos{\left(2 \right)}}{2} + \frac{\sin{\left(2 \right)} \cos^{2}{\left(1 \right)}}{4} - \frac{\sin{\left(1 \right)} \cos{\left(1 \right)} \cos{\left(2 \right)}}{2} + \frac{1}{2} & \text{for}\: n = 2 \\- \frac{n^{2} \cos^{2}{\left(1 \right)} \cos{\left(n \right)}}{n^{3} - 4 n} + \frac{n^{2}}{n^{3} - 4 n} - \frac{2 n \sin{\left(1 \right)} \sin{\left(n \right)} \cos{\left(1 \right)}}{n^{3} - 4 n} + \frac{2 \cos^{2}{\left(1 \right)} \cos{\left(n \right)}}{n^{3} - 4 n} + \frac{2 \sin^{2}{\left(1 \right)} \cos{\left(n \right)}}{n^{3} - 4 n} - \frac{2}{n^{3} - 4 n} & \text{otherwise} \end{cases} 
Piecewise((-1/2 + cos(1)^2*cos(2)/2 - cos(1)^2*sin(2)/4 + sin(1)^2*sin(2)/4 + cos(1)*cos(2)*sin(1)/2 + cos(1)*sin(1)*sin(2)/4, n = -2), (0, n = 0), (1/2 - cos(1)^2*cos(2)/2 - sin(1)^2*sin(2)/4 + cos(1)^2*sin(2)/4 - cos(1)*cos(2)*sin(1)/2 - cos(1)*sin(1)*sin(2)/4, n = 2), (-2/(n^3 - 4*n) + n^2/(n^3 - 4*n) + 2*cos(1)^2*cos(n)/(n^3 - 4*n) + 2*sin(1)^2*cos(n)/(n^3 - 4*n) - n^2*cos(1)^2*cos(n)/(n^3 - 4*n) - 2*n*cos(1)*sin(1)*sin(n)/(n^3 - 4*n), True))

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

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