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Resolución de integrales/ e^x-1/ cos(kx)/ (e^x-1)*cos(kx)

Integral de (e^x-1)*cos(kx) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 2*pi                    
   /                     
  |                      
  |  / x    \            
  |  \E  - 1/*cos(k*x) dx
  |                      
 /                       
 0
∫02π(ex−1)cos⁡(kx) dx\int\limits_{0}^{2 \pi} \left(e^{x} - 1\right) \cos{\left(k x \right)}\, dx 
Integral((E^x - 1)*cos(k*x), (x, 0, 2*pi))
Respuesta (Indefinida) [src] 
  /                                                                                                                   
 |                            //   x      for k = 0\     /                /          x      x         \\              
 | / x    \                   ||                   |     | x              |cos(k*x)*e    k*e *sin(k*x)||             x
 | \E  - 1/*cos(k*x) dx = C - |
∫(ex−1)cos⁡(kx) dx=C+k(−k(kexsin⁡(kx)k2+1+excos⁡(kx)k2+1)+exsin⁡(kx))−{xfor k=0sin⁡(kx)kotherwise+excos⁡(kx)\int \left(e^{x} - 1\right) \cos{\left(k x \right)}\, dx = C + k \left(- k \left(\frac{k e^{x} \sin{\left(k x \right)}}{k^{2} + 1} + \frac{e^{x} \cos{\left(k x \right)}}{k^{2} + 1}\right) + e^{x} \sin{\left(k x \right)}\right) - \begin{cases} x & \text{for}\: k = 0 \\\frac{\sin{\left(k x \right)}}{k} & \text{otherwise} \end{cases} + e^{x} \cos{\left(k x \right)}
Simplificar
Respuesta [src] 
/                                              2*pi                                             
|                                 -1 - 2*pi + e                                        for k = 0
|                                                                                               
|                          2                              2*pi    2  2*pi                       
<    k      sin(2*pi*k)   k *sin(2*pi*k)   k*cos(2*pi*k)*e       k *e    *sin(2*pi*k)           
|- ------ - ----------- - -------------- + ------------------- + --------------------  otherwise
|       3           3              3                   3                     3                  
|  k + k       k + k          k + k               k + k                 k + k                   
\
{−2π−1+e2πfor k=0−k2sin⁡(2πk)k3+k+k2e2πsin⁡(2πk)k3+k+ke2πcos⁡(2πk)k3+k−kk3+k−sin⁡(2πk)k3+kotherwise\begin{cases} - 2 \pi - 1 + e^{2 \pi} & \text{for}\: k = 0 \\- \frac{k^{2} \sin{\left(2 \pi k \right)}}{k^{3} + k} + \frac{k^{2} e^{2 \pi} \sin{\left(2 \pi k \right)}}{k^{3} + k} + \frac{k e^{2 \pi} \cos{\left(2 \pi k \right)}}{k^{3} + k} - \frac{k}{k^{3} + k} - \frac{\sin{\left(2 \pi k \right)}}{k^{3} + k} & \text{otherwise} \end{cases} 
=
= 
/                                              2*pi                                             
|                                 -1 - 2*pi + e                                        for k = 0
|                                                                                               
|                          2                              2*pi    2  2*pi                       
<    k      sin(2*pi*k)   k *sin(2*pi*k)   k*cos(2*pi*k)*e       k *e    *sin(2*pi*k)           
|- ------ - ----------- - -------------- + ------------------- + --------------------  otherwise
|       3           3              3                   3                     3                  
|  k + k       k + k          k + k               k + k                 k + k                   
\
{−2π−1+e2πfor k=0−k2sin⁡(2πk)k3+k+k2e2πsin⁡(2πk)k3+k+ke2πcos⁡(2πk)k3+k−kk3+k−sin⁡(2πk)k3+kotherwise\begin{cases} - 2 \pi - 1 + e^{2 \pi} & \text{for}\: k = 0 \\- \frac{k^{2} \sin{\left(2 \pi k \right)}}{k^{3} + k} + \frac{k^{2} e^{2 \pi} \sin{\left(2 \pi k \right)}}{k^{3} + k} + \frac{k e^{2 \pi} \cos{\left(2 \pi k \right)}}{k^{3} + k} - \frac{k}{k^{3} + k} - \frac{\sin{\left(2 \pi k \right)}}{k^{3} + k} & \text{otherwise} \end{cases} 
Piecewise((-1 - 2*pi + exp(2*pi), k = 0), (-k/(k + k^3) - sin(2*pi*k)/(k + k^3) - k^2*sin(2*pi*k)/(k + k^3) + k*cos(2*pi*k)*exp(2*pi)/(k + k^3) + k^2*exp(2*pi)*sin(2*pi*k)/(k + k^3), True))

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

    © Sr Examen – Калькулятор