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Resolución de integrales/ e^(a*x)/ sin(2*m*x)/e^(a*x)

Integral de sin(2*m*x)/e^(a*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1              
  /              
 |               
 |  sin(2*m*x)   
 |  ---------- dx
 |      a*x      
 |     E         
 |               
/                
0
∫01sin⁡(2mx)eax dx\int\limits_{0}^{1} \frac{\sin{\left(2 m x \right)}}{e^{a x}}\, dx 
Integral(sin((2*m)*x)/E^(a*x), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                      /                   
 |                      |                    
 | sin(2*m*x)           |  -a*x              
 | ---------- dx = C +  | e    *sin(2*m*x) dx
 |     a*x              |                    
 |    E                /                     
 |                                           
/
∫sin⁡(2mx)eax dx=C+∫e−axsin⁡(2mx) dx\int \frac{\sin{\left(2 m x \right)}}{e^{a x}}\, dx = C + \int e^{- a x} \sin{\left(2 m x \right)}\, dx
Simplificar
Respuesta [src] 
/                            0                                                 for Or(And(a = 0, m = 0), And(a = 0, a = -2*I*m, m = 0), And(a = 0, a = 2*I*m, m = 0), And(a = 0, a = -2*I*m, a = 2*I*m, m = 0))                  
|                                                                                                                                                                                                                                
|  2*I*m                        2*I*m      2*I*m                                                                                                                                                                                 
| e     *sin(2*m)   I*cos(2*m)*e        I*e     *sin(2*m)                                                                                                                                                                        
| --------------- + ----------------- - -----------------    for Or(And(a = 0, a = -2*I*m), And(a = -2*I*m, a = 2*I*m), And(a = -2*I*m, m = 0), And(a = 0, a = -2*I*m, a = 2*I*m), And(a = -2*I*m, a = 2*I*m, m = 0), a = -2*I*m)
|        2                  2                  4*m                                                                                                                                                                               
|                                                                                                                                                                                                                                
| -2*I*m                        -2*I*m      -2*I*m
{0for (a=0∧m=0)∨(a=0∧a=−2im∧m=0)∨(a=0∧a=2im∧m=0)∨(a=0∧a=−2im∧a=2im∧m=0)e2imsin⁡(2m)2+ie2imcos⁡(2m)2−ie2imsin⁡(2m)4mfor (a=0∧a=−2im)∨(a=−2im∧a=2im)∨(a=−2im∧m=0)∨(a=0∧a=−2im∧a=2im)∨(a=−2im∧a=2im∧m=0)∨a=−2ime−2imsin⁡(2m)2−ie−2imcos⁡(2m)2+ie−2imsin⁡(2m)4mfor (a=0∧a=2im)∨(a=2im∧m=0)∨a=2im−asin⁡(2m)a2ea+4m2ea−2mcos⁡(2m)a2ea+4m2ea+2ma2+4m2otherwise\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge m = 0\right) \vee \left(a = 0 \wedge a = - 2 i m \wedge m = 0\right) \vee \left(a = 0 \wedge a = 2 i m \wedge m = 0\right) \vee \left(a = 0 \wedge a = - 2 i m \wedge a = 2 i m \wedge m = 0\right) \\\frac{e^{2 i m} \sin{\left(2 m \right)}}{2} + \frac{i e^{2 i m} \cos{\left(2 m \right)}}{2} - \frac{i e^{2 i m} \sin{\left(2 m \right)}}{4 m} & \text{for}\: \left(a = 0 \wedge a = - 2 i m\right) \vee \left(a = - 2 i m \wedge a = 2 i m\right) \vee \left(a = - 2 i m \wedge m = 0\right) \vee \left(a = 0 \wedge a = - 2 i m \wedge a = 2 i m\right) \vee \left(a = - 2 i m \wedge a = 2 i m \wedge m = 0\right) \vee a = - 2 i m \\\frac{e^{- 2 i m} \sin{\left(2 m \right)}}{2} - \frac{i e^{- 2 i m} \cos{\left(2 m \right)}}{2} + \frac{i e^{- 2 i m} \sin{\left(2 m \right)}}{4 m} & \text{for}\: \left(a = 0 \wedge a = 2 i m\right) \vee \left(a = 2 i m \wedge m = 0\right) \vee a = 2 i m \\- \frac{a \sin{\left(2 m \right)}}{a^{2} e^{a} + 4 m^{2} e^{a}} - \frac{2 m \cos{\left(2 m \right)}}{a^{2} e^{a} + 4 m^{2} e^{a}} + \frac{2 m}{a^{2} + 4 m^{2}} & \text{otherwise} \end{cases} 
=
= 
/                            0                                                 for Or(And(a = 0, m = 0), And(a = 0, a = -2*I*m, m = 0), And(a = 0, a = 2*I*m, m = 0), And(a = 0, a = -2*I*m, a = 2*I*m, m = 0))                  
|                                                                                                                                                                                                                                
|  2*I*m                        2*I*m      2*I*m                                                                                                                                                                                 
| e     *sin(2*m)   I*cos(2*m)*e        I*e     *sin(2*m)                                                                                                                                                                        
| --------------- + ----------------- - -----------------    for Or(And(a = 0, a = -2*I*m), And(a = -2*I*m, a = 2*I*m), And(a = -2*I*m, m = 0), And(a = 0, a = -2*I*m, a = 2*I*m), And(a = -2*I*m, a = 2*I*m, m = 0), a = -2*I*m)
|        2                  2                  4*m                                                                                                                                                                               
|                                                                                                                                                                                                                                
| -2*I*m                        -2*I*m      -2*I*m
{0for (a=0∧m=0)∨(a=0∧a=−2im∧m=0)∨(a=0∧a=2im∧m=0)∨(a=0∧a=−2im∧a=2im∧m=0)e2imsin⁡(2m)2+ie2imcos⁡(2m)2−ie2imsin⁡(2m)4mfor (a=0∧a=−2im)∨(a=−2im∧a=2im)∨(a=−2im∧m=0)∨(a=0∧a=−2im∧a=2im)∨(a=−2im∧a=2im∧m=0)∨a=−2ime−2imsin⁡(2m)2−ie−2imcos⁡(2m)2+ie−2imsin⁡(2m)4mfor (a=0∧a=2im)∨(a=2im∧m=0)∨a=2im−asin⁡(2m)a2ea+4m2ea−2mcos⁡(2m)a2ea+4m2ea+2ma2+4m2otherwise\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge m = 0\right) \vee \left(a = 0 \wedge a = - 2 i m \wedge m = 0\right) \vee \left(a = 0 \wedge a = 2 i m \wedge m = 0\right) \vee \left(a = 0 \wedge a = - 2 i m \wedge a = 2 i m \wedge m = 0\right) \\\frac{e^{2 i m} \sin{\left(2 m \right)}}{2} + \frac{i e^{2 i m} \cos{\left(2 m \right)}}{2} - \frac{i e^{2 i m} \sin{\left(2 m \right)}}{4 m} & \text{for}\: \left(a = 0 \wedge a = - 2 i m\right) \vee \left(a = - 2 i m \wedge a = 2 i m\right) \vee \left(a = - 2 i m \wedge m = 0\right) \vee \left(a = 0 \wedge a = - 2 i m \wedge a = 2 i m\right) \vee \left(a = - 2 i m \wedge a = 2 i m \wedge m = 0\right) \vee a = - 2 i m \\\frac{e^{- 2 i m} \sin{\left(2 m \right)}}{2} - \frac{i e^{- 2 i m} \cos{\left(2 m \right)}}{2} + \frac{i e^{- 2 i m} \sin{\left(2 m \right)}}{4 m} & \text{for}\: \left(a = 0 \wedge a = 2 i m\right) \vee \left(a = 2 i m \wedge m = 0\right) \vee a = 2 i m \\- \frac{a \sin{\left(2 m \right)}}{a^{2} e^{a} + 4 m^{2} e^{a}} - \frac{2 m \cos{\left(2 m \right)}}{a^{2} e^{a} + 4 m^{2} e^{a}} + \frac{2 m}{a^{2} + 4 m^{2}} & \text{otherwise} \end{cases} 
Piecewise((0, ((a = 0)∧(m = 0))∨((a = 0)∧(m = 0)∧(a = -2*i*m))∨((a = 0)∧(m = 0)∧(a = 2*i*m))∨((a = 0)∧(m = 0)∧(a = -2*i*m)∧(a = 2*i*m))), (exp(2*i*m)*sin(2*m)/2 + i*cos(2*m)*exp(2*i*m)/2 - i*exp(2*i*m)*sin(2*m)/(4*m), (a = -2*i*m)∨((a = 0)∧(a = -2*i*m))∨((m = 0)∧(a = -2*i*m))∨((a = -2*i*m)∧(a = 2*i*m))∨((a = 0)∧(a = -2*i*m)∧(a = 2*i*m))∨((m = 0)∧(a = -2*i*m)∧(a = 2*i*m))), (exp(-2*i*m)*sin(2*m)/2 - i*cos(2*m)*exp(-2*i*m)/2 + i*exp(-2*i*m)*sin(2*m)/(4*m), (a = 2*i*m)∨((a = 0)∧(a = 2*i*m))∨((m = 0)∧(a = 2*i*m))), (2*m/(a^2 + 4*m^2) - a*sin(2*m)/(a^2*exp(a) + 4*m^2*exp(a)) - 2*m*cos(2*m)/(a^2*exp(a) + 4*m^2*exp(a)), True))

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

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