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

Integral de e^(a*x)*sin(b)*x 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
∫01xeaxsin⁡(b) dx\int\limits_{0}^{1} x e^{a x} \sin{\left(b \right)}\, dx 
Integral((E^(a*x)*sin(b))*x, (x, 0, 1))
Respuesta (Indefinida) [src] 
                          //                        a*x             \
                          ||(-sin(b) + a*x*sin(b))*e          2     |
  /                       ||---------------------------  for a  != 0|
 |                        ||              2                         |
 |  a*x                   ||             a                          |
 | E   *sin(b)*x dx = C + |<                                        |
 |                        ||          2                             |
/                         ||         x *sin(b)                      |
                          ||         ---------            otherwise |
                          ||             2                          |
                          \\                                        /
∫xeaxsin⁡(b) dx=C+{(axsin⁡(b)−sin⁡(b))eaxa2for a2≠0x2sin⁡(b)2otherwise\int x e^{a x} \sin{\left(b \right)}\, dx = C + \begin{cases} \frac{\left(a x \sin{\left(b \right)} - \sin{\left(b \right)}\right) e^{a x}}{a^{2}} & \text{for}\: a^{2} \neq 0 \\\frac{x^{2} \sin{\left(b \right)}}{2} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/                               a                                  
|sin(b)   (-sin(b) + a*sin(b))*e                                   
|------ + -----------------------  for And(a > -oo, a < oo, a != 0)
|   2                 2                                            
<  a                 a                                             
|                                                                  
|             sin(b)                                               
|             ------                          otherwise            
\               2
{(asin⁡(b)−sin⁡(b))eaa2+sin⁡(b)a2for a>−∞∧a<∞∧a≠0sin⁡(b)2otherwise\begin{cases} \frac{\left(a \sin{\left(b \right)} - \sin{\left(b \right)}\right) e^{a}}{a^{2}} + \frac{\sin{\left(b \right)}}{a^{2}} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\\frac{\sin{\left(b \right)}}{2} & \text{otherwise} \end{cases} 
=
= 
/                               a                                  
|sin(b)   (-sin(b) + a*sin(b))*e                                   
|------ + -----------------------  for And(a > -oo, a < oo, a != 0)
|   2                 2                                            
<  a                 a                                             
|                                                                  
|             sin(b)                                               
|             ------                          otherwise            
\               2
{(asin⁡(b)−sin⁡(b))eaa2+sin⁡(b)a2for a>−∞∧a<∞∧a≠0sin⁡(b)2otherwise\begin{cases} \frac{\left(a \sin{\left(b \right)} - \sin{\left(b \right)}\right) e^{a}}{a^{2}} + \frac{\sin{\left(b \right)}}{a^{2}} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\\frac{\sin{\left(b \right)}}{2} & \text{otherwise} \end{cases} 
Piecewise((sin(b)/a^2 + (-sin(b) + a*sin(b))*exp(a)/a^2, (a > -oo)∧(a < oo)∧(Ne(a, 0))), (sin(b)/2, True))

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

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