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Resolución de integrales/ (ax-b)^a

Integral de (ax-b)^a dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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

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

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