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Integral de e^(-x/4)sinnx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
 pi                 
  /                 
 |                  
 |   -x             
 |   ---            
 |    4             
 |  E   *sin(n*x) dx
 |                  
/                   
0                   
$$\int\limits_{0}^{\pi} e^{\frac{\left(-1\right) x}{4}} \sin{\left(n x \right)}\, dx$$
Integral(E^((-x)/4)*sin(n*x), (x, 0, pi))
Respuesta (Indefinida) [src]
                          //   /                            -x       -x         \             \
                          ||   |                            ---      ---        |             |
                          ||   |           -x          /x\   4        4      /x\|             |
                          ||   |           ---   x*cosh|-|*e      x*e   *sinh|-||             |
                          ||   |      /x\   4          \4/                   \4/|          -I |
                          ||-I*|2*cosh|-|*e    + -------------- + --------------|  for n = ---|
  /                       ||   \      \4/              2                2       /           4 |
 |                        ||                                                                  |
 |  -x                    ||  /                            -x       -x         \              |
 |  ---                   ||  |                            ---      ---        |              |
 |   4                    ||  |           -x          /x\   4        4      /x\|              |
 | E   *sin(n*x) dx = C + |<  |           ---   x*cosh|-|*e      x*e   *sinh|-||              |
 |                        ||  |      /x\   4          \4/                   \4/|            I |
/                         ||I*|2*cosh|-|*e    + -------------- + --------------|    for n = - |
                          ||  \      \4/              2                2       /            4 |
                          ||                                                                  |
                          ||               4*sin(n*x)    16*n*cos(n*x)                        |
                          ||           - ------------- - -------------              otherwise |
                          ||                    x    x          x    x                        |
                          ||                    -    -          -    -                        |
                          ||                 2  4    4       2  4    4                        |
                          ||             16*n *e  + e    16*n *e  + e                         |
                          \\                                                                  /
$$\int e^{\frac{\left(-1\right) x}{4}} \sin{\left(n x \right)}\, dx = C + \begin{cases} - i \left(\frac{x e^{- \frac{x}{4}} \sinh{\left(\frac{x}{4} \right)}}{2} + \frac{x e^{- \frac{x}{4}} \cosh{\left(\frac{x}{4} \right)}}{2} + 2 e^{- \frac{x}{4}} \cosh{\left(\frac{x}{4} \right)}\right) & \text{for}\: n = - \frac{i}{4} \\i \left(\frac{x e^{- \frac{x}{4}} \sinh{\left(\frac{x}{4} \right)}}{2} + \frac{x e^{- \frac{x}{4}} \cosh{\left(\frac{x}{4} \right)}}{2} + 2 e^{- \frac{x}{4}} \cosh{\left(\frac{x}{4} \right)}\right) & \text{for}\: n = \frac{i}{4} \\- \frac{16 n \cos{\left(n x \right)}}{16 n^{2} e^{\frac{x}{4}} + e^{\frac{x}{4}}} - \frac{4 \sin{\left(n x \right)}}{16 n^{2} e^{\frac{x}{4}} + e^{\frac{x}{4}}} & \text{otherwise} \end{cases}$$
Respuesta [src]
    4*sin(pi*n)        16*n      16*n*cos(pi*n)
- --------------- + --------- - ---------------
         pi    pi           2          pi    pi
         --    --   1 + 16*n           --    --
      2  4     4                    2  4     4 
  16*n *e   + e                 16*n *e   + e  
$$- \frac{16 n \cos{\left(\pi n \right)}}{16 n^{2} e^{\frac{\pi}{4}} + e^{\frac{\pi}{4}}} + \frac{16 n}{16 n^{2} + 1} - \frac{4 \sin{\left(\pi n \right)}}{16 n^{2} e^{\frac{\pi}{4}} + e^{\frac{\pi}{4}}}$$
=
=
    4*sin(pi*n)        16*n      16*n*cos(pi*n)
- --------------- + --------- - ---------------
         pi    pi           2          pi    pi
         --    --   1 + 16*n           --    --
      2  4     4                    2  4     4 
  16*n *e   + e                 16*n *e   + e  
$$- \frac{16 n \cos{\left(\pi n \right)}}{16 n^{2} e^{\frac{\pi}{4}} + e^{\frac{\pi}{4}}} + \frac{16 n}{16 n^{2} + 1} - \frac{4 \sin{\left(\pi n \right)}}{16 n^{2} e^{\frac{\pi}{4}} + e^{\frac{\pi}{4}}}$$
-4*sin(pi*n)/(16*n^2*exp(pi/4) + exp(pi/4)) + 16*n/(1 + 16*n^2) - 16*n*cos(pi*n)/(16*n^2*exp(pi/4) + exp(pi/4))

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