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Integral de e^(ax)sin(bx) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
  1                 
  /                 
 |                  
 |   a*x            
 |  E   *sin(b*x) dx
 |                  
/                   
0                   
$$\int\limits_{0}^{1} e^{a x} \sin{\left(b x \right)}\, dx$$
Integral(E^(a*x)*sin(b*x), (x, 0, 1))
Respuesta (Indefinida) [src]
                          //                             0                                for And(a = 0, b = 0)\
                          ||                                                                                   |
                          ||   -I*b*x                      -I*b*x                 -I*b*x                       |
                          ||x*e      *sin(b*x)   cos(b*x)*e         I*x*cos(b*x)*e                             |
                          ||------------------ - ---------------- - --------------------      for a = -I*b     |
  /                       ||        2                  2*b                   2                                 |
 |                        ||                                                                                   |
 |  a*x                   ||    I*b*x                      I*b*x                 I*b*x                         |
 | E   *sin(b*x) dx = C + |< x*e     *sin(b*x)   cos(b*x)*e        I*x*cos(b*x)*e                              |
 |                        || ----------------- - --------------- + -------------------         for a = I*b     |
/                         ||         2                 2*b                  2                                  |
                          ||                                                                                   |
                          ||                a*x                        a*x                                     |
                          ||             a*e   *sin(b*x)   b*cos(b*x)*e                                        |
                          ||             --------------- - ---------------                      otherwise      |
                          ||                  2    2            2    2                                         |
                          \\                 a  + b            a  + b                                          /
$$\int e^{a x} \sin{\left(b x \right)}\, dx = C + \begin{cases} 0 & \text{for}\: a = 0 \wedge b = 0 \\\frac{x e^{- i b x} \sin{\left(b x \right)}}{2} - \frac{i x e^{- i b x} \cos{\left(b x \right)}}{2} - \frac{e^{- i b x} \cos{\left(b x \right)}}{2 b} & \text{for}\: a = - i b \\\frac{x e^{i b x} \sin{\left(b x \right)}}{2} + \frac{i x e^{i b x} \cos{\left(b x \right)}}{2} - \frac{e^{i b x} \cos{\left(b x \right)}}{2 b} & \text{for}\: a = i b \\\frac{a e^{a x} \sin{\left(b x \right)}}{a^{2} + b^{2}} - \frac{b e^{a x} \cos{\left(b x \right)}}{a^{2} + b^{2}} & \text{otherwise} \end{cases}$$
Respuesta [src]
/                        0                                        for Or(And(a = 0, b = 0), And(a = 0, a = -I*b, b = 0), And(a = 0, a = I*b, b = 0), And(a = 0, a = -I*b, a = I*b, b = 0))             
|                                                                                                                                                                                                      
|       -I*b                    -I*b           -I*b                                                                                                                                                    
| 1    e    *sin(b)   I*cos(b)*e       cos(b)*e                                                                                                                                                        
|--- + ------------ - -------------- - ------------  for Or(And(a = 0, a = -I*b), And(a = -I*b, a = I*b), And(a = -I*b, b = 0), And(a = 0, a = -I*b, a = I*b), And(a = -I*b, a = I*b, b = 0), a = -I*b)
|2*b        2               2              2*b                                                                                                                                                         
|                                                                                                                                                                                                      
|        I*b                    I*b           I*b                                                                                                                                                      
<  1    e   *sin(b)   I*cos(b)*e      cos(b)*e                                                                                                                                                         
| --- + ----------- + ------------- - -----------                                                for Or(And(a = 0, a = I*b), And(a = I*b, b = 0), a = I*b)                                             
| 2*b        2              2             2*b                                                                                                                                                          
|                                                                                                                                                                                                      
|                    a                    a                                                                                                                                                            
|          b      a*e *sin(b)   b*cos(b)*e                                                                                                                                                             
|       ------- + ----------- - -----------                                                                              otherwise                                                                     
|        2    2      2    2        2    2                                                                                                                                                              
\       a  + b      a  + b        a  + b                                                                                                                                                               
$$\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b \wedge b = 0\right) \\\frac{e^{- i b} \sin{\left(b \right)}}{2} - \frac{i e^{- i b} \cos{\left(b \right)}}{2} + \frac{1}{2 b} - \frac{e^{- i b} \cos{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = - i b\right) \vee \left(a = - i b \wedge a = i b\right) \vee \left(a = - i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b\right) \vee \left(a = - i b \wedge a = i b \wedge b = 0\right) \vee a = - i b \\\frac{e^{i b} \sin{\left(b \right)}}{2} + \frac{i e^{i b} \cos{\left(b \right)}}{2} - \frac{e^{i b} \cos{\left(b \right)}}{2 b} + \frac{1}{2 b} & \text{for}\: \left(a = 0 \wedge a = i b\right) \vee \left(a = i b \wedge b = 0\right) \vee a = i b \\\frac{a e^{a} \sin{\left(b \right)}}{a^{2} + b^{2}} - \frac{b e^{a} \cos{\left(b \right)}}{a^{2} + b^{2}} + \frac{b}{a^{2} + b^{2}} & \text{otherwise} \end{cases}$$
=
=
/                        0                                        for Or(And(a = 0, b = 0), And(a = 0, a = -I*b, b = 0), And(a = 0, a = I*b, b = 0), And(a = 0, a = -I*b, a = I*b, b = 0))             
|                                                                                                                                                                                                      
|       -I*b                    -I*b           -I*b                                                                                                                                                    
| 1    e    *sin(b)   I*cos(b)*e       cos(b)*e                                                                                                                                                        
|--- + ------------ - -------------- - ------------  for Or(And(a = 0, a = -I*b), And(a = -I*b, a = I*b), And(a = -I*b, b = 0), And(a = 0, a = -I*b, a = I*b), And(a = -I*b, a = I*b, b = 0), a = -I*b)
|2*b        2               2              2*b                                                                                                                                                         
|                                                                                                                                                                                                      
|        I*b                    I*b           I*b                                                                                                                                                      
<  1    e   *sin(b)   I*cos(b)*e      cos(b)*e                                                                                                                                                         
| --- + ----------- + ------------- - -----------                                                for Or(And(a = 0, a = I*b), And(a = I*b, b = 0), a = I*b)                                             
| 2*b        2              2             2*b                                                                                                                                                          
|                                                                                                                                                                                                      
|                    a                    a                                                                                                                                                            
|          b      a*e *sin(b)   b*cos(b)*e                                                                                                                                                             
|       ------- + ----------- - -----------                                                                              otherwise                                                                     
|        2    2      2    2        2    2                                                                                                                                                              
\       a  + b      a  + b        a  + b                                                                                                                                                               
$$\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b \wedge b = 0\right) \\\frac{e^{- i b} \sin{\left(b \right)}}{2} - \frac{i e^{- i b} \cos{\left(b \right)}}{2} + \frac{1}{2 b} - \frac{e^{- i b} \cos{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = - i b\right) \vee \left(a = - i b \wedge a = i b\right) \vee \left(a = - i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b\right) \vee \left(a = - i b \wedge a = i b \wedge b = 0\right) \vee a = - i b \\\frac{e^{i b} \sin{\left(b \right)}}{2} + \frac{i e^{i b} \cos{\left(b \right)}}{2} - \frac{e^{i b} \cos{\left(b \right)}}{2 b} + \frac{1}{2 b} & \text{for}\: \left(a = 0 \wedge a = i b\right) \vee \left(a = i b \wedge b = 0\right) \vee a = i b \\\frac{a e^{a} \sin{\left(b \right)}}{a^{2} + b^{2}} - \frac{b e^{a} \cos{\left(b \right)}}{a^{2} + b^{2}} + \frac{b}{a^{2} + b^{2}} & \text{otherwise} \end{cases}$$
Piecewise((0, ((a = 0)∧(b = 0))∨((a = 0)∧(b = 0)∧(a = i*b))∨((a = 0)∧(b = 0)∧(a = -i*b))∨((a = 0)∧(b = 0)∧(a = i*b)∧(a = -i*b))), (1/(2*b) + exp(-i*b)*sin(b)/2 - i*cos(b)*exp(-i*b)/2 - cos(b)*exp(-i*b)/(2*b), (a = -i*b)∨((a = 0)∧(a = -i*b))∨((b = 0)∧(a = -i*b))∨((a = i*b)∧(a = -i*b))∨((a = 0)∧(a = i*b)∧(a = -i*b))∨((b = 0)∧(a = i*b)∧(a = -i*b))), (1/(2*b) + exp(i*b)*sin(b)/2 + i*cos(b)*exp(i*b)/2 - cos(b)*exp(i*b)/(2*b), (a = i*b)∨((a = 0)∧(a = i*b))∨((b = 0)∧(a = i*b))), (b/(a^2 + b^2) + a*exp(a)*sin(b)/(a^2 + b^2) - b*cos(b)*exp(a)/(a^2 + b^2), True))

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