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Resolución de integrales/ (ax+b)/ ln(ax+b)/√(ax+b)

Integral de ln(ax+b)/√(ax+b) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                
  /                
 |                 
 |  log(a*x + b)   
 |  ------------ dx
 |    _________    
 |  \/ a*x + b     
 |                 
/                  
0
$$\int\limits_{0}^{1} \frac{\log{\left(a x + b \right)}}{\sqrt{a x + b}}\, dx$$ 
Integral(log(a*x + b)/sqrt(a*x + b), (x, 0, 1))
Respuesta (Indefinida) [src] 
                         //      _________       _________                         \
  /                      ||- 4*\/ a*x + b  + 2*\/ a*x + b *log(a*x + b)            |
 |                       ||--------------------------------------------  for a != 0|
 | log(a*x + b)          ||                     a                                  |
 | ------------ dx = C + |<                                                        |
 |   _________           ||                  x*log(b)                              |
 | \/ a*x + b            ||                  --------                    otherwise |
 |                       ||                     ___                                |
/                        \\                   \/ b                                 /
$$\int \frac{\log{\left(a x + b \right)}}{\sqrt{a x + b}}\, dx = C + \begin{cases} \frac{2 \sqrt{a x + b} \log{\left(a x + b \right)} - 4 \sqrt{a x + b}}{a} & \text{for}\: a \neq 0 \\\frac{x \log{\left(b \right)}}{\sqrt{b}} & \text{otherwise} \end{cases}$$
Simplificar
Respuesta [src] 
/    ___       ___              _______       _______                                             
|4*\/ b  - 2*\/ b *log(b)   4*\/ a + b  - 2*\/ a + b *log(a + b)                                  
|------------------------ - ------------------------------------  for And(a > -oo, a < oo, a != 0)
|           a                                a                                                    
<                                                                                                 
|                            log(b)                                                               
|                            ------                                          otherwise            
|                              ___                                                                
\                            \/ b
$$\begin{cases} \frac{- 2 \sqrt{b} \log{\left(b \right)} + 4 \sqrt{b}}{a} - \frac{- 2 \sqrt{a + b} \log{\left(a + b \right)} + 4 \sqrt{a + b}}{a} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\\frac{\log{\left(b \right)}}{\sqrt{b}} & \text{otherwise} \end{cases}$$ 
=
= 
/    ___       ___              _______       _______                                             
|4*\/ b  - 2*\/ b *log(b)   4*\/ a + b  - 2*\/ a + b *log(a + b)                                  
|------------------------ - ------------------------------------  for And(a > -oo, a < oo, a != 0)
|           a                                a                                                    
<                                                                                                 
|                            log(b)                                                               
|                            ------                                          otherwise            
|                              ___                                                                
\                            \/ b
$$\begin{cases} \frac{- 2 \sqrt{b} \log{\left(b \right)} + 4 \sqrt{b}}{a} - \frac{- 2 \sqrt{a + b} \log{\left(a + b \right)} + 4 \sqrt{a + b}}{a} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\\frac{\log{\left(b \right)}}{\sqrt{b}} & \text{otherwise} \end{cases}$$ 
Piecewise(((4*sqrt(b) - 2*sqrt(b)*log(b))/a - (4*sqrt(a + b) - 2*sqrt(a + b)*log(a + b))/a, (a > -oo)∧(a < oo)∧(Ne(a, 0))), (log(b)/sqrt(b), True))

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

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