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Resolución de integrales/ X*sinx/ X*sinx*sinkx

Integral de X*sinx*sinkx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
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 |  x*sin(x)*sin(k*x) dx
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∫01xsin⁡(x)sin⁡(kx) dx\int\limits_{0}^{1} x \sin{\left(x \right)} \sin{\left(k x \right)}\, dx 
Integral((x*sin(x))*sin(k*x), (x, 0, 1))
Respuesta (Indefinida) [src] 
                              //                                                  2       2    2       2    2                                                                                 \
                              ||                                               cos (x)   x *cos (x)   x *sin (x)   x*cos(x)*sin(x)                                                            |
                              ||                                               ------- - ---------- - ---------- + ---------------                                                  for k = -1|
                              ||                                                  4          4            4               2                                                                   |
                              ||                                                                                                                                                              |
  /                           ||                                                   2       2    2       2    2                                                                                |
 |                            ||                                                cos (x)   x *cos (x)   x *sin (x)   x*cos(x)*sin(x)                                                           |
 | x*sin(x)*sin(k*x) dx = C + |<                                              - ------- + ---------- + ---------- - ---------------                                                 for k = 1 |
 |                            ||                                                   4          4            4               2                                                                  |
/                             ||                                                                                                                                                              |
                              ||                   2                                                                                      2                      3                            |
                              ||sin(x)*sin(k*x)   k *sin(x)*sin(k*x)   x*cos(x)*sin(k*x)   2*k*cos(x)*cos(k*x)   k*x*cos(k*x)*sin(x)   x*k *cos(x)*sin(k*x)   x*k *cos(k*x)*sin(x)            |
                              ||--------------- + ------------------ - ----------------- + ------------------- + ------------------- + -------------------- - --------------------  otherwise |
                              ||      4      2           4      2             4      2             4      2              4      2              4      2               4      2                |
                              \\ 1 + k  - 2*k       1 + k  - 2*k         1 + k  - 2*k         1 + k  - 2*k          1 + k  - 2*k          1 + k  - 2*k           1 + k  - 2*k                 /
∫xsin⁡(x)sin⁡(kx) dx=C+{−x2sin⁡2(x)4−x2cos⁡2(x)4+xsin⁡(x)cos⁡(x)2+cos⁡2(x)4for k=−1x2sin⁡2(x)4+x2cos⁡2(x)4−xsin⁡(x)cos⁡(x)2−cos⁡2(x)4for k=1−k3xsin⁡(x)cos⁡(kx)k4−2k2+1+k2xsin⁡(kx)cos⁡(x)k4−2k2+1+k2sin⁡(x)sin⁡(kx)k4−2k2+1+kxsin⁡(x)cos⁡(kx)k4−2k2+1+2kcos⁡(x)cos⁡(kx)k4−2k2+1−xsin⁡(kx)cos⁡(x)k4−2k2+1+sin⁡(x)sin⁡(kx)k4−2k2+1otherwise\int x \sin{\left(x \right)} \sin{\left(k x \right)}\, dx = C + \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}\: k = -1 \\\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}\: k = 1 \\- \frac{k^{3} x \sin{\left(x \right)} \cos{\left(k x \right)}}{k^{4} - 2 k^{2} + 1} + \frac{k^{2} x \sin{\left(k x \right)} \cos{\left(x \right)}}{k^{4} - 2 k^{2} + 1} + \frac{k^{2} \sin{\left(x \right)} \sin{\left(k x \right)}}{k^{4} - 2 k^{2} + 1} + \frac{k x \sin{\left(x \right)} \cos{\left(k x \right)}}{k^{4} - 2 k^{2} + 1} + \frac{2 k \cos{\left(x \right)} \cos{\left(k x \right)}}{k^{4} - 2 k^{2} + 1} - \frac{x \sin{\left(k x \right)} \cos{\left(x \right)}}{k^{4} - 2 k^{2} + 1} + \frac{\sin{\left(x \right)} \sin{\left(k x \right)}}{k^{4} - 2 k^{2} + 1} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/                                                                 2                                                                                        
|                                                          1   sin (1)   cos(1)*sin(1)                                                                     
|                                                        - - - ------- + -------------                                                           for k = -1
|                                                          4      4            2                                                                           
|                                                                                                                                                          
|                                                                2                                                                                         
|                                                         1   sin (1)   cos(1)*sin(1)                                                                      
<                                                         - + ------- - -------------                                                            for k = 1 
|                                                         4      4            2                                                                            
|                                                                                                                                                          
|                                                                     2                  2                  3                                              
|       2*k        sin(1)*sin(k)   cos(1)*sin(k)   k*cos(k)*sin(1)   k *cos(1)*sin(k)   k *sin(1)*sin(k)   k *cos(k)*sin(1)   2*k*cos(1)*cos(k)            
|- ------------- + ------------- - ------------- + --------------- + ---------------- + ---------------- - ---------------- + -----------------  otherwise 
|       4      2        4      2        4      2         4      2          4      2           4      2           4      2            4      2              
\  1 + k  - 2*k    1 + k  - 2*k    1 + k  - 2*k     1 + k  - 2*k      1 + k  - 2*k       1 + k  - 2*k       1 + k  - 2*k        1 + k  - 2*k
{−14−sin⁡2(1)4+sin⁡(1)cos⁡(1)2for k=−1−sin⁡(1)cos⁡(1)2+sin⁡2(1)4+14for k=1−k3sin⁡(1)cos⁡(k)k4−2k2+1+k2sin⁡(k)cos⁡(1)k4−2k2+1+k2sin⁡(1)sin⁡(k)k4−2k2+1+ksin⁡(1)cos⁡(k)k4−2k2+1+2kcos⁡(1)cos⁡(k)k4−2k2+1−2kk4−2k2+1−sin⁡(k)cos⁡(1)k4−2k2+1+sin⁡(1)sin⁡(k)k4−2k2+1otherwise\begin{cases} - \frac{1}{4} - \frac{\sin^{2}{\left(1 \right)}}{4} + \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} & \text{for}\: k = -1 \\- \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{\sin^{2}{\left(1 \right)}}{4} + \frac{1}{4} & \text{for}\: k = 1 \\- \frac{k^{3} \sin{\left(1 \right)} \cos{\left(k \right)}}{k^{4} - 2 k^{2} + 1} + \frac{k^{2} \sin{\left(k \right)} \cos{\left(1 \right)}}{k^{4} - 2 k^{2} + 1} + \frac{k^{2} \sin{\left(1 \right)} \sin{\left(k \right)}}{k^{4} - 2 k^{2} + 1} + \frac{k \sin{\left(1 \right)} \cos{\left(k \right)}}{k^{4} - 2 k^{2} + 1} + \frac{2 k \cos{\left(1 \right)} \cos{\left(k \right)}}{k^{4} - 2 k^{2} + 1} - \frac{2 k}{k^{4} - 2 k^{2} + 1} - \frac{\sin{\left(k \right)} \cos{\left(1 \right)}}{k^{4} - 2 k^{2} + 1} + \frac{\sin{\left(1 \right)} \sin{\left(k \right)}}{k^{4} - 2 k^{2} + 1} & \text{otherwise} \end{cases} 
=
= 
/                                                                 2                                                                                        
|                                                          1   sin (1)   cos(1)*sin(1)                                                                     
|                                                        - - - ------- + -------------                                                           for k = -1
|                                                          4      4            2                                                                           
|                                                                                                                                                          
|                                                                2                                                                                         
|                                                         1   sin (1)   cos(1)*sin(1)                                                                      
<                                                         - + ------- - -------------                                                            for k = 1 
|                                                         4      4            2                                                                            
|                                                                                                                                                          
|                                                                     2                  2                  3                                              
|       2*k        sin(1)*sin(k)   cos(1)*sin(k)   k*cos(k)*sin(1)   k *cos(1)*sin(k)   k *sin(1)*sin(k)   k *cos(k)*sin(1)   2*k*cos(1)*cos(k)            
|- ------------- + ------------- - ------------- + --------------- + ---------------- + ---------------- - ---------------- + -----------------  otherwise 
|       4      2        4      2        4      2         4      2          4      2           4      2           4      2            4      2              
\  1 + k  - 2*k    1 + k  - 2*k    1 + k  - 2*k     1 + k  - 2*k      1 + k  - 2*k       1 + k  - 2*k       1 + k  - 2*k        1 + k  - 2*k
{−14−sin⁡2(1)4+sin⁡(1)cos⁡(1)2for k=−1−sin⁡(1)cos⁡(1)2+sin⁡2(1)4+14for k=1−k3sin⁡(1)cos⁡(k)k4−2k2+1+k2sin⁡(k)cos⁡(1)k4−2k2+1+k2sin⁡(1)sin⁡(k)k4−2k2+1+ksin⁡(1)cos⁡(k)k4−2k2+1+2kcos⁡(1)cos⁡(k)k4−2k2+1−2kk4−2k2+1−sin⁡(k)cos⁡(1)k4−2k2+1+sin⁡(1)sin⁡(k)k4−2k2+1otherwise\begin{cases} - \frac{1}{4} - \frac{\sin^{2}{\left(1 \right)}}{4} + \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} & \text{for}\: k = -1 \\- \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{\sin^{2}{\left(1 \right)}}{4} + \frac{1}{4} & \text{for}\: k = 1 \\- \frac{k^{3} \sin{\left(1 \right)} \cos{\left(k \right)}}{k^{4} - 2 k^{2} + 1} + \frac{k^{2} \sin{\left(k \right)} \cos{\left(1 \right)}}{k^{4} - 2 k^{2} + 1} + \frac{k^{2} \sin{\left(1 \right)} \sin{\left(k \right)}}{k^{4} - 2 k^{2} + 1} + \frac{k \sin{\left(1 \right)} \cos{\left(k \right)}}{k^{4} - 2 k^{2} + 1} + \frac{2 k \cos{\left(1 \right)} \cos{\left(k \right)}}{k^{4} - 2 k^{2} + 1} - \frac{2 k}{k^{4} - 2 k^{2} + 1} - \frac{\sin{\left(k \right)} \cos{\left(1 \right)}}{k^{4} - 2 k^{2} + 1} + \frac{\sin{\left(1 \right)} \sin{\left(k \right)}}{k^{4} - 2 k^{2} + 1} & \text{otherwise} \end{cases} 
Piecewise((-1/4 - sin(1)^2/4 + cos(1)*sin(1)/2, k = -1), (1/4 + sin(1)^2/4 - cos(1)*sin(1)/2, k = 1), (-2*k/(1 + k^4 - 2*k^2) + sin(1)*sin(k)/(1 + k^4 - 2*k^2) - cos(1)*sin(k)/(1 + k^4 - 2*k^2) + k*cos(k)*sin(1)/(1 + k^4 - 2*k^2) + k^2*cos(1)*sin(k)/(1 + k^4 - 2*k^2) + k^2*sin(1)*sin(k)/(1 + k^4 - 2*k^2) - k^3*cos(k)*sin(1)/(1 + k^4 - 2*k^2) + 2*k*cos(1)*cos(k)/(1 + k^4 - 2*k^2), True))

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

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