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Resolución de integrales/ e^(a*x)/ e^(a*x)*cos(b*x)

Integral de e^(a*x)*cos(b*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                 
  /                 
 |                  
 |   a*x            
 |  E   *cos(b*x) dx
 |                  
/                   
0
∫01eaxcos⁡(bx) dx\int\limits_{0}^{1} e^{a x} \cos{\left(b x \right)}\, dx 
Integral(E^(a*x)*cos(b*x), (x, 0, 1))
Respuesta (Indefinida) [src] 
                          //                             x                                for And(a = 0, b = 0)\
                          ||                                                                                   |
                          ||            -I*b*x    -I*b*x                 -I*b*x                                |
                          ||x*cos(b*x)*e         e      *sin(b*x)   I*x*e      *sin(b*x)                       |
                          ||------------------ + ---------------- + --------------------      for a = -I*b     |
  /                       ||        2                  2*b                   2                                 |
 |                        ||                                                                                   |
 |  a*x                   ||             I*b*x    I*b*x                 I*b*x                                  |
 | E   *cos(b*x) dx = C + |< x*cos(b*x)*e        e     *sin(b*x)   I*x*e     *sin(b*x)                         |
 |                        || ----------------- + --------------- - -------------------         for a = I*b     |
/                         ||         2                 2*b                  2                                  |
                          ||                                                                                   |
                          ||                         a*x      a*x                                              |
                          ||             a*cos(b*x)*e      b*e   *sin(b*x)                                     |
                          ||             --------------- + ---------------                      otherwise      |
                          ||                  2    2            2    2                                         |
                          \\                 a  + b            a  + b                                          /
∫eaxcos⁡(bx) dx=C+{xfor a=0∧b=0ixe−ibxsin⁡(bx)2+xe−ibxcos⁡(bx)2+e−ibxsin⁡(bx)2bfor a=−ib−ixeibxsin⁡(bx)2+xeibxcos⁡(bx)2+eibxsin⁡(bx)2bfor a=ibaeaxcos⁡(bx)a2+b2+beaxsin⁡(bx)a2+b2otherwise\int e^{a x} \cos{\left(b x \right)}\, dx = C + \begin{cases} x & \text{for}\: a = 0 \wedge b = 0 \\\frac{i x e^{- i b x} \sin{\left(b x \right)}}{2} + \frac{x e^{- i b x} \cos{\left(b x \right)}}{2} + \frac{e^{- i b x} \sin{\left(b x \right)}}{2 b} & \text{for}\: a = - i b \\- \frac{i x e^{i b x} \sin{\left(b x \right)}}{2} + \frac{x e^{i b x} \cos{\left(b x \right)}}{2} + \frac{e^{i b x} \sin{\left(b x \right)}}{2 b} & \text{for}\: a = i b \\\frac{a e^{a x} \cos{\left(b x \right)}}{a^{2} + b^{2}} + \frac{b e^{a x} \sin{\left(b x \right)}}{a^{2} + b^{2}} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/                         1                                         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                                                                                                                                                    
|cos(b)*e        I    I*e    *sin(b)   I*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         2*b         2               2*b                                                                                                                                                          
|                                                                                                                                                                                                        
|               I*b      I*b                    I*b                                                                                                                                                      
<  I    cos(b)*e      I*e   *sin(b)   I*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                                                                                                                                                                   
|            a      a*cos(b)*e    b*e *sin(b)                                                                                                                                                            
|       - ------- + ----------- + -----------                                                                              otherwise                                                                     
|          2    2      2    2        2    2                                                                                                                                                              
\         a  + b      a  + b        a  + b
{1for (a=0∧b=0)∨(a=0∧a=−ib∧b=0)∨(a=0∧a=ib∧b=0)∨(a=0∧a=−ib∧a=ib∧b=0)ie−ibsin⁡(b)2+e−ibcos⁡(b)2−i2b+ie−ibcos⁡(b)2bfor (a=0∧a=−ib)∨(a=−ib∧a=ib)∨(a=−ib∧b=0)∨(a=0∧a=−ib∧a=ib)∨(a=−ib∧a=ib∧b=0)∨a=−ib−ieibsin⁡(b)2+eibcos⁡(b)2−ieibcos⁡(b)2b+i2bfor (a=0∧a=ib)∨(a=ib∧b=0)∨a=ibaeacos⁡(b)a2+b2−aa2+b2+beasin⁡(b)a2+b2otherwise\begin{cases} 1 & \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{i e^{- i b} \sin{\left(b \right)}}{2} + \frac{e^{- i b} \cos{\left(b \right)}}{2} - \frac{i}{2 b} + \frac{i 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{i e^{i b} \sin{\left(b \right)}}{2} + \frac{e^{i b} \cos{\left(b \right)}}{2} - \frac{i e^{i b} \cos{\left(b \right)}}{2 b} + \frac{i}{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} \cos{\left(b \right)}}{a^{2} + b^{2}} - \frac{a}{a^{2} + b^{2}} + \frac{b e^{a} \sin{\left(b \right)}}{a^{2} + b^{2}} & \text{otherwise} \end{cases} 
=
= 
/                         1                                         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                                                                                                                                                    
|cos(b)*e        I    I*e    *sin(b)   I*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         2*b         2               2*b                                                                                                                                                          
|                                                                                                                                                                                                        
|               I*b      I*b                    I*b                                                                                                                                                      
<  I    cos(b)*e      I*e   *sin(b)   I*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                                                                                                                                                                   
|            a      a*cos(b)*e    b*e *sin(b)                                                                                                                                                            
|       - ------- + ----------- + -----------                                                                              otherwise                                                                     
|          2    2      2    2        2    2                                                                                                                                                              
\         a  + b      a  + b        a  + b
{1for (a=0∧b=0)∨(a=0∧a=−ib∧b=0)∨(a=0∧a=ib∧b=0)∨(a=0∧a=−ib∧a=ib∧b=0)ie−ibsin⁡(b)2+e−ibcos⁡(b)2−i2b+ie−ibcos⁡(b)2bfor (a=0∧a=−ib)∨(a=−ib∧a=ib)∨(a=−ib∧b=0)∨(a=0∧a=−ib∧a=ib)∨(a=−ib∧a=ib∧b=0)∨a=−ib−ieibsin⁡(b)2+eibcos⁡(b)2−ieibcos⁡(b)2b+i2bfor (a=0∧a=ib)∨(a=ib∧b=0)∨a=ibaeacos⁡(b)a2+b2−aa2+b2+beasin⁡(b)a2+b2otherwise\begin{cases} 1 & \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{i e^{- i b} \sin{\left(b \right)}}{2} + \frac{e^{- i b} \cos{\left(b \right)}}{2} - \frac{i}{2 b} + \frac{i 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{i e^{i b} \sin{\left(b \right)}}{2} + \frac{e^{i b} \cos{\left(b \right)}}{2} - \frac{i e^{i b} \cos{\left(b \right)}}{2 b} + \frac{i}{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} \cos{\left(b \right)}}{a^{2} + b^{2}} - \frac{a}{a^{2} + b^{2}} + \frac{b e^{a} \sin{\left(b \right)}}{a^{2} + b^{2}} & \text{otherwise} \end{cases} 
Piecewise((1, ((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))), (cos(b)*exp(-i*b)/2 - i/(2*b) + i*exp(-i*b)*sin(b)/2 + i*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))), (i/(2*b) + cos(b)*exp(i*b)/2 - i*exp(i*b)*sin(b)/2 - i*cos(b)*exp(i*b)/(2*b), (a = i*b)∨((a = 0)∧(a = i*b))∨((b = 0)∧(a = i*b))), (-a/(a^2 + b^2) + a*cos(b)*exp(a)/(a^2 + b^2) + b*exp(a)*sin(b)/(a^2 + b^2), True))

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

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