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Resolución de integrales/ sin3x/ (sin3x+sin3πx)sinπkx/5

Integral de (sin3x+sin3πx)sinπkx/5 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  5                                        
  /                                        
 |                                         
 |  (sin(3*x) + sin(3*pi*x))*sin(pi*k*x)   
 |  ------------------------------------ dx
 |                   5                     
 |                                         
/                                          
0
∫05(sin⁡(3x)+sin⁡(3πx))sin⁡(xπk)5 dx\int\limits_{0}^{5} \frac{\left(\sin{\left(3 x \right)} + \sin{\left(3 \pi x \right)}\right) \sin{\left(x \pi k \right)}}{5}\, dx 
Integral(((sin(3*x) + sin((3*pi)*x))*sin((pi*k)*x))/5, (x, 0, 5))
Respuesta [src] 
/                                                                                                             3                             2                                                                                                                          
|                                                                                                         3*pi             3*pi         2*pi *sin(15)                                                                                                                  
|                                                                                                   - ------------- + ------------- - -----------------                                                                                                     for k = -3 
|                                                                                                                 3               3     /            3\                                                                                                                
|                                                                                                     -6*pi + 6*pi    -6*pi + 6*pi    5*\-6*pi + 6*pi /                                                                                                                
|                                                                                                                                                                                                                                                                      
|                                                                                                                             3             2                                                                                                                          
|                                                                                                          3*pi           3*pi          2*pi *sin(15)                                                                                                                  
|                                                                                                   - ------------- + ------------- + -----------------                                                                                                      for k = 3 
|                                                                                                                 3               3     /            3\                                                                                                                
|                                                                                                     -6*pi + 6*pi    -6*pi + 6*pi    5*\-6*pi + 6*pi /                                                                                                                
|                                                                                                                                                                                                                                                                      
|                                                                        2            2           2    2           2    2                                            2                                                                                                 
|                                                                   3*cos (15)   3*sin (15)   3*pi *cos (15)   3*pi *sin (15)    2*pi*sin(15)    cos(15)*sin(15)   pi *cos(15)*sin(15)                                                                              -3 
<                                                                   ---------- + ---------- - -------------- - -------------- - -------------- - --------------- + -------------------                                                                      for k = ---
|                                                                            2            2              2                2       /         2\      /         2\        /         2\                                                                                 pi
|                                                                   -6 + 6*pi    -6 + 6*pi      -6 + 6*pi        -6 + 6*pi      5*\-6 + 6*pi /    5*\-6 + 6*pi /      5*\-6 + 6*pi /                                                                                   
|                                                                                                                                                                                                                                                                      
|                                                                         2            2           2    2           2    2                                            2                                                                                                
|                                                                    3*cos (15)   3*sin (15)   3*pi *cos (15)   3*pi *sin (15)   cos(15)*sin(15)    2*pi*sin(15)    pi *cos(15)*sin(15)                                                                             3  
|                                                                  - ---------- - ---------- + -------------- + -------------- + --------------- + -------------- - -------------------                                                                     for k = -- 
|                                                                             2            2              2                2        /         2\     /         2\        /         2\                                                                               pi 
|                                                                    -6 + 6*pi    -6 + 6*pi      -6 + 6*pi        -6 + 6*pi       5*\-6 + 6*pi /   5*\-6 + 6*pi /      5*\-6 + 6*pi /                                                                                  
|                                                                                                                                                                                                                                                                      
|                                                                                                  2  2                                 2  3                                         2                                         2                                       
|             27*sin(5*pi*k)                      27*pi*cos(15)*sin(5*pi*k)                    3*pi *k *sin(5*pi*k)                   pi *k *cos(5*pi*k)*sin(15)               3*pi*k *cos(15)*sin(5*pi*k)               9*k*pi *cos(5*pi*k)*sin(15)                   
|--------------------------------------- - --------------------------------------- - --------------------------------------- - --------------------------------------- + --------------------------------------- + ---------------------------------------   otherwise 
|  /          3  4         2       3  2\     /          3  4         2       3  2\     /          3  4         2       3  2\     /          3  4         2       3  2\     /          3  4         2       3  2\     /          3  4         2       3  2\             
\5*\81*pi + pi *k  - 9*pi*k  - 9*pi *k /   5*\81*pi + pi *k  - 9*pi*k  - 9*pi *k /   5*\81*pi + pi *k  - 9*pi*k  - 9*pi *k /   5*\81*pi + pi *k  - 9*pi*k  - 9*pi *k /   5*\81*pi + pi *k  - 9*pi*k  - 9*pi *k /   5*\81*pi + pi *k  - 9*pi*k  - 9*pi *k /
{−3π3−6π+6π3−2π2sin⁡(15)5(−6π+6π3)+3π−6π+6π3for k=−3−3π−6π+6π3+2π2sin⁡(15)5(−6π+6π3)+3π3−6π+6π3for k=3−3π2cos⁡2(15)−6+6π2−3π2sin⁡2(15)−6+6π2+π2sin⁡(15)cos⁡(15)5(−6+6π2)−2πsin⁡(15)5(−6+6π2)−sin⁡(15)cos⁡(15)5(−6+6π2)+3sin⁡2(15)−6+6π2+3cos⁡2(15)−6+6π2for k=−3π−3cos⁡2(15)−6+6π2−3sin⁡2(15)−6+6π2+sin⁡(15)cos⁡(15)5(−6+6π2)+2πsin⁡(15)5(−6+6π2)−π2sin⁡(15)cos⁡(15)5(−6+6π2)+3π2sin⁡2(15)−6+6π2+3π2cos⁡2(15)−6+6π2for k=3π−π2k3sin⁡(15)cos⁡(5πk)5(π3k4−9π3k2−9πk2+81π)−3π2k2sin⁡(5πk)5(π3k4−9π3k2−9πk2+81π)+3πk2sin⁡(5πk)cos⁡(15)5(π3k4−9π3k2−9πk2+81π)+9π2ksin⁡(15)cos⁡(5πk)5(π3k4−9π3k2−9πk2+81π)+27sin⁡(5πk)5(π3k4−9π3k2−9πk2+81π)−27πsin⁡(5πk)cos⁡(15)5(π3k4−9π3k2−9πk2+81π)otherwise\begin{cases} - \frac{3 \pi^{3}}{- 6 \pi + 6 \pi^{3}} - \frac{2 \pi^{2} \sin{\left(15 \right)}}{5 \left(- 6 \pi + 6 \pi^{3}\right)} + \frac{3 \pi}{- 6 \pi + 6 \pi^{3}} & \text{for}\: k = -3 \\- \frac{3 \pi}{- 6 \pi + 6 \pi^{3}} + \frac{2 \pi^{2} \sin{\left(15 \right)}}{5 \left(- 6 \pi + 6 \pi^{3}\right)} + \frac{3 \pi^{3}}{- 6 \pi + 6 \pi^{3}} & \text{for}\: k = 3 \\- \frac{3 \pi^{2} \cos^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} - \frac{3 \pi^{2} \sin^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} + \frac{\pi^{2} \sin{\left(15 \right)} \cos{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} - \frac{2 \pi \sin{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} - \frac{\sin{\left(15 \right)} \cos{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} + \frac{3 \sin^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} + \frac{3 \cos^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} & \text{for}\: k = - \frac{3}{\pi} \\- \frac{3 \cos^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} - \frac{3 \sin^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} + \frac{\sin{\left(15 \right)} \cos{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} + \frac{2 \pi \sin{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} - \frac{\pi^{2} \sin{\left(15 \right)} \cos{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} + \frac{3 \pi^{2} \sin^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} + \frac{3 \pi^{2} \cos^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} & \text{for}\: k = \frac{3}{\pi} \\- \frac{\pi^{2} k^{3} \sin{\left(15 \right)} \cos{\left(5 \pi k \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} - \frac{3 \pi^{2} k^{2} \sin{\left(5 \pi k \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} + \frac{3 \pi k^{2} \sin{\left(5 \pi k \right)} \cos{\left(15 \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} + \frac{9 \pi^{2} k \sin{\left(15 \right)} \cos{\left(5 \pi k \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} + \frac{27 \sin{\left(5 \pi k \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} - \frac{27 \pi \sin{\left(5 \pi k \right)} \cos{\left(15 \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} & \text{otherwise} \end{cases} 
=
= 
/                                                                                                             3                             2                                                                                                                          
|                                                                                                         3*pi             3*pi         2*pi *sin(15)                                                                                                                  
|                                                                                                   - ------------- + ------------- - -----------------                                                                                                     for k = -3 
|                                                                                                                 3               3     /            3\                                                                                                                
|                                                                                                     -6*pi + 6*pi    -6*pi + 6*pi    5*\-6*pi + 6*pi /                                                                                                                
|                                                                                                                                                                                                                                                                      
|                                                                                                                             3             2                                                                                                                          
|                                                                                                          3*pi           3*pi          2*pi *sin(15)                                                                                                                  
|                                                                                                   - ------------- + ------------- + -----------------                                                                                                      for k = 3 
|                                                                                                                 3               3     /            3\                                                                                                                
|                                                                                                     -6*pi + 6*pi    -6*pi + 6*pi    5*\-6*pi + 6*pi /                                                                                                                
|                                                                                                                                                                                                                                                                      
|                                                                        2            2           2    2           2    2                                            2                                                                                                 
|                                                                   3*cos (15)   3*sin (15)   3*pi *cos (15)   3*pi *sin (15)    2*pi*sin(15)    cos(15)*sin(15)   pi *cos(15)*sin(15)                                                                              -3 
<                                                                   ---------- + ---------- - -------------- - -------------- - -------------- - --------------- + -------------------                                                                      for k = ---
|                                                                            2            2              2                2       /         2\      /         2\        /         2\                                                                                 pi
|                                                                   -6 + 6*pi    -6 + 6*pi      -6 + 6*pi        -6 + 6*pi      5*\-6 + 6*pi /    5*\-6 + 6*pi /      5*\-6 + 6*pi /                                                                                   
|                                                                                                                                                                                                                                                                      
|                                                                         2            2           2    2           2    2                                            2                                                                                                
|                                                                    3*cos (15)   3*sin (15)   3*pi *cos (15)   3*pi *sin (15)   cos(15)*sin(15)    2*pi*sin(15)    pi *cos(15)*sin(15)                                                                             3  
|                                                                  - ---------- - ---------- + -------------- + -------------- + --------------- + -------------- - -------------------                                                                     for k = -- 
|                                                                             2            2              2                2        /         2\     /         2\        /         2\                                                                               pi 
|                                                                    -6 + 6*pi    -6 + 6*pi      -6 + 6*pi        -6 + 6*pi       5*\-6 + 6*pi /   5*\-6 + 6*pi /      5*\-6 + 6*pi /                                                                                  
|                                                                                                                                                                                                                                                                      
|                                                                                                  2  2                                 2  3                                         2                                         2                                       
|             27*sin(5*pi*k)                      27*pi*cos(15)*sin(5*pi*k)                    3*pi *k *sin(5*pi*k)                   pi *k *cos(5*pi*k)*sin(15)               3*pi*k *cos(15)*sin(5*pi*k)               9*k*pi *cos(5*pi*k)*sin(15)                   
|--------------------------------------- - --------------------------------------- - --------------------------------------- - --------------------------------------- + --------------------------------------- + ---------------------------------------   otherwise 
|  /          3  4         2       3  2\     /          3  4         2       3  2\     /          3  4         2       3  2\     /          3  4         2       3  2\     /          3  4         2       3  2\     /          3  4         2       3  2\             
\5*\81*pi + pi *k  - 9*pi*k  - 9*pi *k /   5*\81*pi + pi *k  - 9*pi*k  - 9*pi *k /   5*\81*pi + pi *k  - 9*pi*k  - 9*pi *k /   5*\81*pi + pi *k  - 9*pi*k  - 9*pi *k /   5*\81*pi + pi *k  - 9*pi*k  - 9*pi *k /   5*\81*pi + pi *k  - 9*pi*k  - 9*pi *k /
{−3π3−6π+6π3−2π2sin⁡(15)5(−6π+6π3)+3π−6π+6π3for k=−3−3π−6π+6π3+2π2sin⁡(15)5(−6π+6π3)+3π3−6π+6π3for k=3−3π2cos⁡2(15)−6+6π2−3π2sin⁡2(15)−6+6π2+π2sin⁡(15)cos⁡(15)5(−6+6π2)−2πsin⁡(15)5(−6+6π2)−sin⁡(15)cos⁡(15)5(−6+6π2)+3sin⁡2(15)−6+6π2+3cos⁡2(15)−6+6π2for k=−3π−3cos⁡2(15)−6+6π2−3sin⁡2(15)−6+6π2+sin⁡(15)cos⁡(15)5(−6+6π2)+2πsin⁡(15)5(−6+6π2)−π2sin⁡(15)cos⁡(15)5(−6+6π2)+3π2sin⁡2(15)−6+6π2+3π2cos⁡2(15)−6+6π2for k=3π−π2k3sin⁡(15)cos⁡(5πk)5(π3k4−9π3k2−9πk2+81π)−3π2k2sin⁡(5πk)5(π3k4−9π3k2−9πk2+81π)+3πk2sin⁡(5πk)cos⁡(15)5(π3k4−9π3k2−9πk2+81π)+9π2ksin⁡(15)cos⁡(5πk)5(π3k4−9π3k2−9πk2+81π)+27sin⁡(5πk)5(π3k4−9π3k2−9πk2+81π)−27πsin⁡(5πk)cos⁡(15)5(π3k4−9π3k2−9πk2+81π)otherwise\begin{cases} - \frac{3 \pi^{3}}{- 6 \pi + 6 \pi^{3}} - \frac{2 \pi^{2} \sin{\left(15 \right)}}{5 \left(- 6 \pi + 6 \pi^{3}\right)} + \frac{3 \pi}{- 6 \pi + 6 \pi^{3}} & \text{for}\: k = -3 \\- \frac{3 \pi}{- 6 \pi + 6 \pi^{3}} + \frac{2 \pi^{2} \sin{\left(15 \right)}}{5 \left(- 6 \pi + 6 \pi^{3}\right)} + \frac{3 \pi^{3}}{- 6 \pi + 6 \pi^{3}} & \text{for}\: k = 3 \\- \frac{3 \pi^{2} \cos^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} - \frac{3 \pi^{2} \sin^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} + \frac{\pi^{2} \sin{\left(15 \right)} \cos{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} - \frac{2 \pi \sin{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} - \frac{\sin{\left(15 \right)} \cos{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} + \frac{3 \sin^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} + \frac{3 \cos^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} & \text{for}\: k = - \frac{3}{\pi} \\- \frac{3 \cos^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} - \frac{3 \sin^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} + \frac{\sin{\left(15 \right)} \cos{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} + \frac{2 \pi \sin{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} - \frac{\pi^{2} \sin{\left(15 \right)} \cos{\left(15 \right)}}{5 \left(-6 + 6 \pi^{2}\right)} + \frac{3 \pi^{2} \sin^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} + \frac{3 \pi^{2} \cos^{2}{\left(15 \right)}}{-6 + 6 \pi^{2}} & \text{for}\: k = \frac{3}{\pi} \\- \frac{\pi^{2} k^{3} \sin{\left(15 \right)} \cos{\left(5 \pi k \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} - \frac{3 \pi^{2} k^{2} \sin{\left(5 \pi k \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} + \frac{3 \pi k^{2} \sin{\left(5 \pi k \right)} \cos{\left(15 \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} + \frac{9 \pi^{2} k \sin{\left(15 \right)} \cos{\left(5 \pi k \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} + \frac{27 \sin{\left(5 \pi k \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} - \frac{27 \pi \sin{\left(5 \pi k \right)} \cos{\left(15 \right)}}{5 \left(\pi^{3} k^{4} - 9 \pi^{3} k^{2} - 9 \pi k^{2} + 81 \pi\right)} & \text{otherwise} \end{cases} 
Piecewise((-3*pi^3/(-6*pi + 6*pi^3) + 3*pi/(-6*pi + 6*pi^3) - 2*pi^2*sin(15)/(5*(-6*pi + 6*pi^3)), k = -3), (-3*pi/(-6*pi + 6*pi^3) + 3*pi^3/(-6*pi + 6*pi^3) + 2*pi^2*sin(15)/(5*(-6*pi + 6*pi^3)), k = 3), (3*cos(15)^2/(-6 + 6*pi^2) + 3*sin(15)^2/(-6 + 6*pi^2) - 3*pi^2*cos(15)^2/(-6 + 6*pi^2) - 3*pi^2*sin(15)^2/(-6 + 6*pi^2) - 2*pi*sin(15)/(5*(-6 + 6*pi^2)) - cos(15)*sin(15)/(5*(-6 + 6*pi^2)) + pi^2*cos(15)*sin(15)/(5*(-6 + 6*pi^2)), k = -3/pi), (-3*cos(15)^2/(-6 + 6*pi^2) - 3*sin(15)^2/(-6 + 6*pi^2) + 3*pi^2*cos(15)^2/(-6 + 6*pi^2) + 3*pi^2*sin(15)^2/(-6 + 6*pi^2) + cos(15)*sin(15)/(5*(-6 + 6*pi^2)) + 2*pi*sin(15)/(5*(-6 + 6*pi^2)) - pi^2*cos(15)*sin(15)/(5*(-6 + 6*pi^2)), k = 3/pi), (27*sin(5*pi*k)/(5*(81*pi + pi^3*k^4 - 9*pi*k^2 - 9*pi^3*k^2)) - 27*pi*cos(15)*sin(5*pi*k)/(5*(81*pi + pi^3*k^4 - 9*pi*k^2 - 9*pi^3*k^2)) - 3*pi^2*k^2*sin(5*pi*k)/(5*(81*pi + pi^3*k^4 - 9*pi*k^2 - 9*pi^3*k^2)) - pi^2*k^3*cos(5*pi*k)*sin(15)/(5*(81*pi + pi^3*k^4 - 9*pi*k^2 - 9*pi^3*k^2)) + 3*pi*k^2*cos(15)*sin(5*pi*k)/(5*(81*pi + pi^3*k^4 - 9*pi*k^2 - 9*pi^3*k^2)) + 9*k*pi^2*cos(5*pi*k)*sin(15)/(5*(81*pi + pi^3*k^4 - 9*pi*k^2 - 9*pi^3*k^2)), True))

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

    © Sr Examen – Калькулятор