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Resolución de integrales/ log(arccos(ax))/sqrt(1-(ax)^2)

Integral de log(arccos(ax))/sqrt(1-(ax)^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                   
  /                   
 |                    
 |   log(acos(a*x))   
 |  --------------- dx
 |     ____________   
 |    /          2    
 |  \/  1 - (a*x)     
 |                    
/                     
0
∫01log⁡(acos⁡(ax))1−(ax)2 dx\int\limits_{0}^{1} \frac{\log{\left(\operatorname{acos}{\left(a x \right)} \right)}}{\sqrt{1 - \left(a x\right)^{2}}}\, dx 
Integral(log(acos(a*x))/sqrt(1 - (a*x)^2), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                         //-(-acos(a*x) + acos(a*x)*log(acos(a*x)))             \
 |                          ||-----------------------------------------  for a != 0|
 |  log(acos(a*x))          ||                    a                                |
 | --------------- dx = C + |<                                                     |
 |    ____________          ||                     /pi\                            |
 |   /          2           ||                x*log|--|                  otherwise |
 | \/  1 - (a*x)            \\                     \2 /                            /
 |                                                                                  
/
∫log⁡(acos⁡(ax))1−(ax)2 dx=C+{−log⁡(acos⁡(ax))acos⁡(ax)−acos⁡(ax)afor a≠0xlog⁡(π2)otherwise\int \frac{\log{\left(\operatorname{acos}{\left(a x \right)} \right)}}{\sqrt{1 - \left(a x\right)^{2}}}\, dx = C + \begin{cases} - \frac{\log{\left(\operatorname{acos}{\left(a x \right)} \right)} \operatorname{acos}{\left(a x \right)} - \operatorname{acos}{\left(a x \right)}}{a} & \text{for}\: a \neq 0 \\x \log{\left(\frac{\pi}{2} \right)} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/                      /pi\                                                         
|                pi*log|--|                                                         
|acos(a)    pi         \2 /   acos(a)*log(acos(a))                                  
|------- - --- + ---------- - --------------------  for And(a > -oo, a < oo, a != 0)
<   a      2*a      2*a                a                                            
|                                                                                   
|                        /pi\                                                       
|                     log|--|                                  otherwise            
\                        \2 /
{−log⁡(acos⁡(a))acos⁡(a)a+acos⁡(a)a−π2a+πlog⁡(π2)2afor a>−∞∧a<∞∧a≠0log⁡(π2)otherwise\begin{cases} - \frac{\log{\left(\operatorname{acos}{\left(a \right)} \right)} \operatorname{acos}{\left(a \right)}}{a} + \frac{\operatorname{acos}{\left(a \right)}}{a} - \frac{\pi}{2 a} + \frac{\pi \log{\left(\frac{\pi}{2} \right)}}{2 a} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\\log{\left(\frac{\pi}{2} \right)} & \text{otherwise} \end{cases} 
=
= 
/                      /pi\                                                         
|                pi*log|--|                                                         
|acos(a)    pi         \2 /   acos(a)*log(acos(a))                                  
|------- - --- + ---------- - --------------------  for And(a > -oo, a < oo, a != 0)
<   a      2*a      2*a                a                                            
|                                                                                   
|                        /pi\                                                       
|                     log|--|                                  otherwise            
\                        \2 /
{−log⁡(acos⁡(a))acos⁡(a)a+acos⁡(a)a−π2a+πlog⁡(π2)2afor a>−∞∧a<∞∧a≠0log⁡(π2)otherwise\begin{cases} - \frac{\log{\left(\operatorname{acos}{\left(a \right)} \right)} \operatorname{acos}{\left(a \right)}}{a} + \frac{\operatorname{acos}{\left(a \right)}}{a} - \frac{\pi}{2 a} + \frac{\pi \log{\left(\frac{\pi}{2} \right)}}{2 a} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\\log{\left(\frac{\pi}{2} \right)} & \text{otherwise} \end{cases} 
Piecewise((acos(a)/a - pi/(2*a) + pi*log(pi/2)/(2*a) - acos(a)*log(acos(a))/a, (a > -oo)∧(a < oo)∧(Ne(a, 0))), (log(pi/2), True))

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

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