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Resolución de integrales/ exp((1/b)*ln(a+bx))

Integral de exp((1/b)*ln(a+bx)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                 
  /                 
 |                  
 |   log(a + b*x)   
 |   ------------   
 |        b         
 |  e             dx
 |                  
/                   
0
$$\int\limits_{0}^{1} e^{\frac{\log{\left(a + b x \right)}}{b}}\, dx$$ 
Integral(exp(log(a + b*x)/b), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                          
 |                        //   log(a + b*x)        log(a + b*x)             \
 |  log(a + b*x)          ||   ------------        ------------             |
 |  ------------          ||        b                   b                   |
 |       b                ||a*e               b*x*e                         |
 | e             dx = C + |<--------------- + -----------------  for b != -1|
 |                        ||     1 + b              1 + b                   |
/                         ||                                                |
                          ||            -log(x - a)               otherwise |
                          \\                                                /
$$\int e^{\frac{\log{\left(a + b x \right)}}{b}}\, dx = C + \begin{cases} \frac{a e^{\frac{\log{\left(a + b x \right)}}{b}}}{b + 1} + \frac{b x e^{\frac{\log{\left(a + b x \right)}}{b}}}{b + 1} & \text{for}\: b \neq -1 \\- \log{\left(- a + x \right)} & \text{otherwise} \end{cases}$$
Simplificar
Respuesta [src] 
/   log(a + b)      log(a + b)      log(a)                                   
|   ----------      ----------      ------                                   
|       b               b             b                                      
|a*e             b*e             a*e                                         
<------------- + ------------- - ---------  for And(b > -oo, b < oo, b != -1)
|    1 + b           1 + b         1 + b                                     
|                                                                            
|          -log(1 - a) + log(-a)                        otherwise            
\
$$\begin{cases} - \frac{a e^{\frac{\log{\left(a \right)}}{b}}}{b + 1} + \frac{a e^{\frac{\log{\left(a + b \right)}}{b}}}{b + 1} + \frac{b e^{\frac{\log{\left(a + b \right)}}{b}}}{b + 1} & \text{for}\: b > -\infty \wedge b < \infty \wedge b \neq -1 \\\log{\left(- a \right)} - \log{\left(1 - a \right)} & \text{otherwise} \end{cases}$$ 
=
= 
/   log(a + b)      log(a + b)      log(a)                                   
|   ----------      ----------      ------                                   
|       b               b             b                                      
|a*e             b*e             a*e                                         
<------------- + ------------- - ---------  for And(b > -oo, b < oo, b != -1)
|    1 + b           1 + b         1 + b                                     
|                                                                            
|          -log(1 - a) + log(-a)                        otherwise            
\
$$\begin{cases} - \frac{a e^{\frac{\log{\left(a \right)}}{b}}}{b + 1} + \frac{a e^{\frac{\log{\left(a + b \right)}}{b}}}{b + 1} + \frac{b e^{\frac{\log{\left(a + b \right)}}{b}}}{b + 1} & \text{for}\: b > -\infty \wedge b < \infty \wedge b \neq -1 \\\log{\left(- a \right)} - \log{\left(1 - a \right)} & \text{otherwise} \end{cases}$$ 
Piecewise((a*exp(log(a + b)/b)/(1 + b) + b*exp(log(a + b)/b)/(1 + b) - a*exp(log(a)/b)/(1 + b), (b > -oo)∧(b < oo)∧(Ne(b, -1))), (-log(1 - a) + log(-a), True))

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

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