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Resolución de integrales/ e^(ax)/ e^(ax)sinmxdx

Integral de e^(ax)sinmxdx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                 
  /                 
 |                  
 |   a*x            
 |  E   *sin(m*x) dx
 |                  
/                   
0
∫01eaxsin⁡(mx) dx\int\limits_{0}^{1} e^{a x} \sin{\left(m x \right)}\, dx 
Integral(E^(a*x)*sin(m*x), (x, 0, 1))
Respuesta (Indefinida) [src] 
                          //                              0                                 for And(a = 0, m = 0)\
                          ||                                                                                     |
                          ||   -I*m*x               -I*m*x                          -I*m*x                       |
                          ||x*e      *sin(m*x)   I*e      *sin(m*x)   I*x*cos(m*x)*e                             |
                          ||------------------ + ------------------ - --------------------      for a = -I*m     |
  /                       ||        2                   2*m                    2                                 |
 |                        ||                                                                                     |
 |  a*x                   ||    I*m*x                          I*m*x      I*m*x                                  |
 | E   *sin(m*x) dx = C + |< x*e     *sin(m*x)   I*x*cos(m*x)*e        I*e     *sin(m*x)                         |
 |                        || ----------------- + ------------------- - -----------------         for a = I*m     |
/                         ||         2                    2                   2*m                                |
                          ||                                                                                     |
                          ||                 a*x                        a*x                                      |
                          ||              a*e   *sin(m*x)   m*cos(m*x)*e                                         |
                          ||              --------------- - ---------------                       otherwise      |
                          ||                   2    2            2    2                                          |
                          \\                  a  + m            a  + m                                           /
∫eaxsin⁡(mx) dx=C+{0for a=0∧m=0xe−imxsin⁡(mx)2−ixe−imxcos⁡(mx)2+ie−imxsin⁡(mx)2mfor a=−imxeimxsin⁡(mx)2+ixeimxcos⁡(mx)2−ieimxsin⁡(mx)2mfor a=imaeaxsin⁡(mx)a2+m2−meaxcos⁡(mx)a2+m2otherwise\int e^{a x} \sin{\left(m x \right)}\, dx = C + \begin{cases} 0 & \text{for}\: a = 0 \wedge m = 0 \\\frac{x e^{- i m x} \sin{\left(m x \right)}}{2} - \frac{i x e^{- i m x} \cos{\left(m x \right)}}{2} + \frac{i e^{- i m x} \sin{\left(m x \right)}}{2 m} & \text{for}\: a = - i m \\\frac{x e^{i m x} \sin{\left(m x \right)}}{2} + \frac{i x e^{i m x} \cos{\left(m x \right)}}{2} - \frac{i e^{i m x} \sin{\left(m x \right)}}{2 m} & \text{for}\: a = i m \\\frac{a e^{a x} \sin{\left(m x \right)}}{a^{2} + m^{2}} - \frac{m e^{a x} \cos{\left(m x \right)}}{a^{2} + m^{2}} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/                        0                                        for Or(And(a = 0, m = 0), And(a = 0, a = -I*m, m = 0), And(a = 0, a = I*m, m = 0), And(a = 0, a = -I*m, a = I*m, m = 0))             
|                                                                                                                                                                                                      
|       -I*m                    -I*m           -I*m                                                                                                                                                    
| 1    e    *sin(m)   I*cos(m)*e       cos(m)*e                                                                                                                                                        
|--- + ------------ - -------------- - ------------  for Or(And(a = 0, a = -I*m), And(a = -I*m, a = I*m), And(a = -I*m, m = 0), And(a = 0, a = -I*m, a = I*m), And(a = -I*m, a = I*m, m = 0), a = -I*m)
|2*m        2               2              2*m                                                                                                                                                         
|                                                                                                                                                                                                      
|        I*m                    I*m           I*m                                                                                                                                                      
<  1    e   *sin(m)   I*cos(m)*e      cos(m)*e                                                                                                                                                         
| --- + ----------- + ------------- - -----------                                                for Or(And(a = 0, a = I*m), And(a = I*m, m = 0), a = I*m)                                             
| 2*m        2              2             2*m                                                                                                                                                          
|                                                                                                                                                                                                      
|                    a                    a                                                                                                                                                            
|          m      a*e *sin(m)   m*cos(m)*e                                                                                                                                                             
|       ------- + ----------- - -----------                                                                              otherwise                                                                     
|        2    2      2    2        2    2                                                                                                                                                              
\       a  + m      a  + m        a  + m
{0for (a=0∧m=0)∨(a=0∧a=−im∧m=0)∨(a=0∧a=im∧m=0)∨(a=0∧a=−im∧a=im∧m=0)e−imsin⁡(m)2−ie−imcos⁡(m)2+12m−e−imcos⁡(m)2mfor (a=0∧a=−im)∨(a=−im∧a=im)∨(a=−im∧m=0)∨(a=0∧a=−im∧a=im)∨(a=−im∧a=im∧m=0)∨a=−imeimsin⁡(m)2+ieimcos⁡(m)2−eimcos⁡(m)2m+12mfor (a=0∧a=im)∨(a=im∧m=0)∨a=imaeasin⁡(m)a2+m2−meacos⁡(m)a2+m2+ma2+m2otherwise\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge m = 0\right) \vee \left(a = 0 \wedge a = - i m \wedge m = 0\right) \vee \left(a = 0 \wedge a = i m \wedge m = 0\right) \vee \left(a = 0 \wedge a = - i m \wedge a = i m \wedge m = 0\right) \\\frac{e^{- i m} \sin{\left(m \right)}}{2} - \frac{i e^{- i m} \cos{\left(m \right)}}{2} + \frac{1}{2 m} - \frac{e^{- i m} \cos{\left(m \right)}}{2 m} & \text{for}\: \left(a = 0 \wedge a = - i m\right) \vee \left(a = - i m \wedge a = i m\right) \vee \left(a = - i m \wedge m = 0\right) \vee \left(a = 0 \wedge a = - i m \wedge a = i m\right) \vee \left(a = - i m \wedge a = i m \wedge m = 0\right) \vee a = - i m \\\frac{e^{i m} \sin{\left(m \right)}}{2} + \frac{i e^{i m} \cos{\left(m \right)}}{2} - \frac{e^{i m} \cos{\left(m \right)}}{2 m} + \frac{1}{2 m} & \text{for}\: \left(a = 0 \wedge a = i m\right) \vee \left(a = i m \wedge m = 0\right) \vee a = i m \\\frac{a e^{a} \sin{\left(m \right)}}{a^{2} + m^{2}} - \frac{m e^{a} \cos{\left(m \right)}}{a^{2} + m^{2}} + \frac{m}{a^{2} + m^{2}} & \text{otherwise} \end{cases} 
=
= 
/                        0                                        for Or(And(a = 0, m = 0), And(a = 0, a = -I*m, m = 0), And(a = 0, a = I*m, m = 0), And(a = 0, a = -I*m, a = I*m, m = 0))             
|                                                                                                                                                                                                      
|       -I*m                    -I*m           -I*m                                                                                                                                                    
| 1    e    *sin(m)   I*cos(m)*e       cos(m)*e                                                                                                                                                        
|--- + ------------ - -------------- - ------------  for Or(And(a = 0, a = -I*m), And(a = -I*m, a = I*m), And(a = -I*m, m = 0), And(a = 0, a = -I*m, a = I*m), And(a = -I*m, a = I*m, m = 0), a = -I*m)
|2*m        2               2              2*m                                                                                                                                                         
|                                                                                                                                                                                                      
|        I*m                    I*m           I*m                                                                                                                                                      
<  1    e   *sin(m)   I*cos(m)*e      cos(m)*e                                                                                                                                                         
| --- + ----------- + ------------- - -----------                                                for Or(And(a = 0, a = I*m), And(a = I*m, m = 0), a = I*m)                                             
| 2*m        2              2             2*m                                                                                                                                                          
|                                                                                                                                                                                                      
|                    a                    a                                                                                                                                                            
|          m      a*e *sin(m)   m*cos(m)*e                                                                                                                                                             
|       ------- + ----------- - -----------                                                                              otherwise                                                                     
|        2    2      2    2        2    2                                                                                                                                                              
\       a  + m      a  + m        a  + m
{0for (a=0∧m=0)∨(a=0∧a=−im∧m=0)∨(a=0∧a=im∧m=0)∨(a=0∧a=−im∧a=im∧m=0)e−imsin⁡(m)2−ie−imcos⁡(m)2+12m−e−imcos⁡(m)2mfor (a=0∧a=−im)∨(a=−im∧a=im)∨(a=−im∧m=0)∨(a=0∧a=−im∧a=im)∨(a=−im∧a=im∧m=0)∨a=−imeimsin⁡(m)2+ieimcos⁡(m)2−eimcos⁡(m)2m+12mfor (a=0∧a=im)∨(a=im∧m=0)∨a=imaeasin⁡(m)a2+m2−meacos⁡(m)a2+m2+ma2+m2otherwise\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge m = 0\right) \vee \left(a = 0 \wedge a = - i m \wedge m = 0\right) \vee \left(a = 0 \wedge a = i m \wedge m = 0\right) \vee \left(a = 0 \wedge a = - i m \wedge a = i m \wedge m = 0\right) \\\frac{e^{- i m} \sin{\left(m \right)}}{2} - \frac{i e^{- i m} \cos{\left(m \right)}}{2} + \frac{1}{2 m} - \frac{e^{- i m} \cos{\left(m \right)}}{2 m} & \text{for}\: \left(a = 0 \wedge a = - i m\right) \vee \left(a = - i m \wedge a = i m\right) \vee \left(a = - i m \wedge m = 0\right) \vee \left(a = 0 \wedge a = - i m \wedge a = i m\right) \vee \left(a = - i m \wedge a = i m \wedge m = 0\right) \vee a = - i m \\\frac{e^{i m} \sin{\left(m \right)}}{2} + \frac{i e^{i m} \cos{\left(m \right)}}{2} - \frac{e^{i m} \cos{\left(m \right)}}{2 m} + \frac{1}{2 m} & \text{for}\: \left(a = 0 \wedge a = i m\right) \vee \left(a = i m \wedge m = 0\right) \vee a = i m \\\frac{a e^{a} \sin{\left(m \right)}}{a^{2} + m^{2}} - \frac{m e^{a} \cos{\left(m \right)}}{a^{2} + m^{2}} + \frac{m}{a^{2} + m^{2}} & \text{otherwise} \end{cases} 
Piecewise((0, ((a = 0)∧(m = 0))∨((a = 0)∧(m = 0)∧(a = i*m))∨((a = 0)∧(m = 0)∧(a = -i*m))∨((a = 0)∧(m = 0)∧(a = i*m)∧(a = -i*m))), (1/(2*m) + exp(-i*m)*sin(m)/2 - i*cos(m)*exp(-i*m)/2 - cos(m)*exp(-i*m)/(2*m), (a = -i*m)∨((a = 0)∧(a = -i*m))∨((m = 0)∧(a = -i*m))∨((a = i*m)∧(a = -i*m))∨((a = 0)∧(a = i*m)∧(a = -i*m))∨((m = 0)∧(a = i*m)∧(a = -i*m))), (1/(2*m) + exp(i*m)*sin(m)/2 + i*cos(m)*exp(i*m)/2 - cos(m)*exp(i*m)/(2*m), (a = i*m)∨((a = 0)∧(a = i*m))∨((m = 0)∧(a = i*m))), (m/(a^2 + m^2) + a*exp(a)*sin(m)/(a^2 + m^2) - m*cos(m)*exp(a)/(a^2 + m^2), True))

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

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