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Resolución de integrales/ (-x-1)/ (-x-1)*cos(pi/3*(1/2+k)x)

Integral de (-x-1)*cos(pi/3*(1/2+k)x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  3                                
  /                                
 |                                 
 |              /pi            \   
 |  (-x - 1)*cos|--*(1/2 + k)*x| dx
 |              \3             /   
 |                                 
/                                  
0
∫03(−x−1)cos⁡(xπ3(k+12)) dx\int\limits_{0}^{3} \left(- x - 1\right) \cos{\left(x \frac{\pi}{3} \left(k + \frac{1}{2}\right) \right)}\, dx 
Integral((-x - 1)*cos(((pi/3)*(1/2 + k))*x), (x, 0, 3))
Respuesta (Indefinida) [src] 
                                                                                                                                                 ///     /pi*x   pi*k*x\                                    \
                                         //       /pi*x   pi*k*x\               /pi*x   pi*k*x\                /pi*x   pi*k*x\               \   |||6*sin|---- + ------|                                    |
                                         || 36*cos|---- + ------|     6*pi*x*sin|---- + ------|   12*pi*k*x*sin|---- + ------|               |   |||     \ 6       3   /                                    |
  /                                      ||       \ 6       3   /               \ 6       3   /                \ 6       3   /               |   ||<--------------------  for k != -1/2  for pi + 2*pi*k = 0|
 |                                       ||------------------------ + ------------------------- + ----------------------------  for k != -1/2|   |||    pi + 2*pi*k                                         |
 |             /pi            \          ||  2         2       2  2      2         2       2  2       2         2       2  2                 |   |||                                                        |
 | (-x - 1)*cos|--*(1/2 + k)*x| dx = C - |
∫(−x−1)cos⁡(xπ3(k+12)) dx=C−{12πkxsin⁡(πkx3+πx6)4π2k2+4π2k+π2+6πxsin⁡(πkx3+πx6)4π2k2+4π2k+π2+36cos⁡(πkx3+πx6)4π2k2+4π2k+π2for k≠−12x22otherwise−{{6sin⁡(πkx3+πx6)2πk+πfor k≠−12xotherwisefor 2πk+π=06sin⁡(πkx3+πx6)2πk+πotherwise\int \left(- x - 1\right) \cos{\left(x \frac{\pi}{3} \left(k + \frac{1}{2}\right) \right)}\, dx = C - \begin{cases} \frac{12 \pi k x \sin{\left(\frac{\pi k x}{3} + \frac{\pi x}{6} \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} + \frac{6 \pi x \sin{\left(\frac{\pi k x}{3} + \frac{\pi x}{6} \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} + \frac{36 \cos{\left(\frac{\pi k x}{3} + \frac{\pi x}{6} \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} & \text{for}\: k \neq - \frac{1}{2} \\\frac{x^{2}}{2} & \text{otherwise} \end{cases} - \begin{cases} \begin{cases} \frac{6 \sin{\left(\frac{\pi k x}{3} + \frac{\pi x}{6} \right)}}{2 \pi k + \pi} & \text{for}\: k \neq - \frac{1}{2} \\x & \text{otherwise} \end{cases} & \text{for}\: 2 \pi k + \pi = 0 \\\frac{6 \sin{\left(\frac{\pi k x}{3} + \frac{\pi x}{6} \right)}}{2 \pi k + \pi} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/           36              6*cos(pi*k)         36*sin(pi*k)             18*pi*cos(pi*k)           36*pi*k*cos(pi*k)                                         
|------------------------ - ----------- + ------------------------ - ------------------------ - ------------------------  for And(k > -oo, k < oo, k != -1/2)
|  2         2       2  2   pi + 2*pi*k     2         2       2  2     2         2       2  2     2         2       2  2                                     
|pi  + 4*k*pi  + 4*pi *k                  pi  + 4*k*pi  + 4*pi *k    pi  + 4*k*pi  + 4*pi *k    pi  + 4*k*pi  + 4*pi *k                                      
|                                                                                                                                                            
<                                                         -15/2                                                                      for k = -1/2            
|                                                                                                                                                            
|                                                     9   6*cos(pi*k)                                                                                        
|                                                   - - - -----------                                                                  otherwise             
|                                                     2   pi + 2*pi*k                                                                                        
\
{−36πkcos⁡(πk)4π2k2+4π2k+π2+36sin⁡(πk)4π2k2+4π2k+π2−18πcos⁡(πk)4π2k2+4π2k+π2+364π2k2+4π2k+π2−6cos⁡(πk)2πk+πfor k>−∞∧k<∞∧k≠−12−152for k=−12−92−6cos⁡(πk)2πk+πotherwise\begin{cases} - \frac{36 \pi k \cos{\left(\pi k \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} + \frac{36 \sin{\left(\pi k \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} - \frac{18 \pi \cos{\left(\pi k \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} + \frac{36}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} - \frac{6 \cos{\left(\pi k \right)}}{2 \pi k + \pi} & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq - \frac{1}{2} \\- \frac{15}{2} & \text{for}\: k = - \frac{1}{2} \\- \frac{9}{2} - \frac{6 \cos{\left(\pi k \right)}}{2 \pi k + \pi} & \text{otherwise} \end{cases} 
=
= 
/           36              6*cos(pi*k)         36*sin(pi*k)             18*pi*cos(pi*k)           36*pi*k*cos(pi*k)                                         
|------------------------ - ----------- + ------------------------ - ------------------------ - ------------------------  for And(k > -oo, k < oo, k != -1/2)
|  2         2       2  2   pi + 2*pi*k     2         2       2  2     2         2       2  2     2         2       2  2                                     
|pi  + 4*k*pi  + 4*pi *k                  pi  + 4*k*pi  + 4*pi *k    pi  + 4*k*pi  + 4*pi *k    pi  + 4*k*pi  + 4*pi *k                                      
|                                                                                                                                                            
<                                                         -15/2                                                                      for k = -1/2            
|                                                                                                                                                            
|                                                     9   6*cos(pi*k)                                                                                        
|                                                   - - - -----------                                                                  otherwise             
|                                                     2   pi + 2*pi*k                                                                                        
\
{−36πkcos⁡(πk)4π2k2+4π2k+π2+36sin⁡(πk)4π2k2+4π2k+π2−18πcos⁡(πk)4π2k2+4π2k+π2+364π2k2+4π2k+π2−6cos⁡(πk)2πk+πfor k>−∞∧k<∞∧k≠−12−152for k=−12−92−6cos⁡(πk)2πk+πotherwise\begin{cases} - \frac{36 \pi k \cos{\left(\pi k \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} + \frac{36 \sin{\left(\pi k \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} - \frac{18 \pi \cos{\left(\pi k \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} + \frac{36}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} - \frac{6 \cos{\left(\pi k \right)}}{2 \pi k + \pi} & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq - \frac{1}{2} \\- \frac{15}{2} & \text{for}\: k = - \frac{1}{2} \\- \frac{9}{2} - \frac{6 \cos{\left(\pi k \right)}}{2 \pi k + \pi} & \text{otherwise} \end{cases} 
Piecewise((36/(pi^2 + 4*k*pi^2 + 4*pi^2*k^2) - 6*cos(pi*k)/(pi + 2*pi*k) + 36*sin(pi*k)/(pi^2 + 4*k*pi^2 + 4*pi^2*k^2) - 18*pi*cos(pi*k)/(pi^2 + 4*k*pi^2 + 4*pi^2*k^2) - 36*pi*k*cos(pi*k)/(pi^2 + 4*k*pi^2 + 4*pi^2*k^2), (k > -oo)∧(k < oo)∧(Ne(k, -1/2))), (-15/2, k = -1/2), (-9/2 - 6*cos(pi*k)/(pi + 2*pi*k), True))

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

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