Sr Examen

Otras calculadoras

Resolución de integrales/ exp(ax)/ exp(ax)(sin(bx))

Integral de exp(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
∫01eaxsin⁡(bx) dx\int\limits_{0}^{1} e^{a x} \sin{\left(b x \right)}\, dx 
Integral(exp(a*x)*sin(b*x), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                                          //                         /cos(b*x)   x*sin(b*x)                                                \
                                                          ||                         |-------- + ----------  for b != 0                                    |
                                                          ||                         |    2          b                                                     |
                                                          ||                         |   b                                                                 |
                                                          ||                         <                                                            for a = 0|
                                                          ||                         |          2                                                          |
                                                          ||                         |         x                                                           |
                                                          ||                         |         --            otherwise                                     |
                                                          ||                         \         2                                                           |
                                                          ||                                                                                               |
                                                          ||/                             x                                for And(a = 0, b = 0)           |
  /                       // x    for a = 0\              |||                                                                                              |
 |                        ||               |              |||            -I*b*x    -I*b*x                 -I*b*x                                           |
 |  a*x                   || a*x           |              |||x*cos(b*x)*e         e      *sin(b*x)   I*x*e      *sin(b*x)                                  |
 | e   *sin(b*x) dx = C + |
∫eaxsin⁡(bx) dx=C−b({{xsin⁡(bx)b+cos⁡(bx)b2for b≠0x22otherwisefor a=0{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+b2otherwiseaotherwise)+({xfor a=0eaxaotherwise)sin⁡(bx)\int e^{a x} \sin{\left(b x \right)}\, dx = C - b \left(\begin{cases} \begin{cases} \frac{x \sin{\left(b x \right)}}{b} + \frac{\cos{\left(b x \right)}}{b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases} & \text{for}\: a = 0 \\\frac{\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}}{a} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} x & \text{for}\: a = 0 \\\frac{e^{a x}}{a} & \text{otherwise} \end{cases}\right) \sin{\left(b x \right)}
Simplificar
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
{0for (a=0∧b=0)∨(a=0∧a=−ib∧b=0)∨(a=0∧a=ib∧b=0)∨(a=0∧a=−ib∧a=ib∧b=0)e−ibsin⁡(b)2−ie−ibcos⁡(b)2+12b−e−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=−ibeibsin⁡(b)2+ieibcos⁡(b)2−eibcos⁡(b)2b+12bfor (a=0∧a=ib)∨(a=ib∧b=0)∨a=ibaeasin⁡(b)a2+b2−beacos⁡(b)a2+b2+ba2+b2otherwise\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
{0for (a=0∧b=0)∨(a=0∧a=−ib∧b=0)∨(a=0∧a=ib∧b=0)∨(a=0∧a=−ib∧a=ib∧b=0)e−ibsin⁡(b)2−ie−ibcos⁡(b)2+12b−e−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=−ibeibsin⁡(b)2+ieibcos⁡(b)2−eibcos⁡(b)2b+12bfor (a=0∧a=ib)∨(a=ib∧b=0)∨a=ibaeasin⁡(b)a2+b2−beacos⁡(b)a2+b2+ba2+b2otherwise\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.

    © Sr Examen – Калькулятор