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Resolución de integrales/ cosbx/ e^(-ax)/ e^(-ax)cosbx

Integral de e^(-ax)cosbx 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
∫01e−axcos⁡(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        I*x*e     *sin(b*x)   I*cos(b*x)*e                              |
                           || ----------------- - ------------------- - -----------------        for a = -I*b     |
  /                        ||         2                    2                   2*b                                |
 |                         ||                                                                                     |
 |  -a*x                   ||            -I*b*x        -I*b*x                        -I*b*x                       |
 | E    *cos(b*x) dx = C + |
∫e−axcos⁡(bx) dx=C+{xfor a=0∧b=0−ixeibxsin⁡(bx)2+xeibxcos⁡(bx)2−ieibxcos⁡(bx)2bfor a=−ibixe−ibxsin⁡(bx)2+xe−ibxcos⁡(bx)2+ie−ibxcos⁡(bx)2bfor a=ib−acos⁡(bx)a2eax+b2eax+bsin⁡(bx)a2eax+b2eaxotherwise\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{i e^{i b x} \cos{\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{i e^{- i b x} \cos{\left(b x \right)}}{2 b} & \text{for}\: a = i b \\- \frac{a \cos{\left(b x \right)}}{a^{2} e^{a x} + b^{2} e^{a x}} + \frac{b \sin{\left(b x \right)}}{a^{2} e^{a x} + b^{2} e^{a x}} & \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                                                                                                                                                      
|  I    cos(b)*e      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*b        2              2              2*b                                                                                                                                                           
|                                                                                                                                                                                                        
|        -I*b            -I*b                    -I*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)−ieibsin⁡(b)2+eibcos⁡(b)2−ieibcos⁡(b)2b+i2bfor (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=−ibie−ibsin⁡(b)2+e−ibcos⁡(b)2−i2b+ie−ibcos⁡(b)2bfor (a=0∧a=ib)∨(a=ib∧b=0)∨a=ib−acos⁡(b)a2ea+b2ea+aa2+b2+bsin⁡(b)a2ea+b2eaotherwise\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 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 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}{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 b = 0\right) \vee a = i b \\- \frac{a \cos{\left(b \right)}}{a^{2} e^{a} + b^{2} e^{a}} + \frac{a}{a^{2} + b^{2}} + \frac{b \sin{\left(b \right)}}{a^{2} e^{a} + b^{2} e^{a}} & \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                                                                                                                                                      
|  I    cos(b)*e      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*b        2              2              2*b                                                                                                                                                           
|                                                                                                                                                                                                        
|        -I*b            -I*b                    -I*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)−ieibsin⁡(b)2+eibcos⁡(b)2−ieibcos⁡(b)2b+i2bfor (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=−ibie−ibsin⁡(b)2+e−ibcos⁡(b)2−i2b+ie−ibcos⁡(b)2bfor (a=0∧a=ib)∨(a=ib∧b=0)∨a=ib−acos⁡(b)a2ea+b2ea+aa2+b2+bsin⁡(b)a2ea+b2eaotherwise\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 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 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}{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 b = 0\right) \vee a = i b \\- \frac{a \cos{\left(b \right)}}{a^{2} e^{a} + b^{2} e^{a}} + \frac{a}{a^{2} + b^{2}} + \frac{b \sin{\left(b \right)}}{a^{2} e^{a} + b^{2} e^{a}} & \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))), (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 = i*b)∧(a = -i*b))∨((a = 0)∧(a = i*b)∧(a = -i*b))∨((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/(a^2 + b^2) + b*sin(b)/(a^2*exp(a) + b^2*exp(a)) - a*cos(b)/(a^2*exp(a) + b^2*exp(a)), True))

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

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