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Resolución de integrales/ x^2+1/ 2*x^2/ 1/2*x/ 1/sqrt/ 1/sqrt(c-1/2*x^2+1/6*a*x3)

Integral de 1/sqrt(c-1/2*x^2+1/6*a*x3) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                        
  /                        
 |                         
 |           1             
 |  -------------------- dx
 |       _______________   
 |      /      2           
 |     /      x    a       
 |    /   c - -- + -*x3    
 |  \/        2    6       
 |                         
/                          
0
$$\int\limits_{0}^{1} \frac{1}{\sqrt{\frac{a}{6} x_{3} + \left(c - \frac{x^{2}}{2}\right)}}\, dx$$ 
Integral(1/(sqrt(c - x^2/2 + (a/6)*x3)), (x, 0, 1))
Solución detallada

  1. Vuelva a escribir el integrando:

  2. La integral del producto de una función por una constante es la constante por la integral de esta función:

      PieceweseRule(subfunctions=[(ConstantTimesRule(constant=1/sqrt(a*x3 + 6*c), other=1/sqrt(-3*x**2/(a*x3 + 6*c) + 1), substep=URule(u_var=_u, u_func=sqrt(3)*x*sqrt(1/(a*x3 + 6*c)), constant=sqrt(a*x3/3 + 2*c), substep=ConstantTimesRule(constant=sqrt(a*x3/3 + 2*c), other=1/sqrt(1 - _u**2), substep=ArcsinRule(context=1/sqrt(1 - _u**2), symbol=_u), context=sqrt(a*x3/3 + 2*c)/sqrt(1 - _u**2), symbol=x), context=1/sqrt(-3*x**2/(a*x3 + 6*c) + 1), symbol=x), context=1/sqrt(a*x3 + 6*c - 3*x**2), symbol=x), a*x3 + 6*c > 0)], context=1/sqrt(a*x3 + 6*c - 3*x**2), symbol=x)

    Por lo tanto, el resultado es:

  3. Ahora simplificar:

  4. Añadimos la constante de integración:

Respuesta:

Respuesta (Indefinida) [src] 
  /                                    //    ____________     /            ____________\                    \
 |                                     ||   /       a*x3      |    ___    /     1      |                    |
 |          1                      ___ ||  /  2*c + ---- *asin|x*\/ 3 *  /  ---------- |                    |
 | -------------------- dx = C + \/ 6 *|<\/          3        \        \/   6*c + a*x3 /                    |
 |      _______________                ||-----------------------------------------------  for 6*c + a*x3 > 0|
 |     /      2                        ||                   ____________                                    |
 |    /      x    a                    \\                 \/ 6*c + a*x3                                     /
 |   /   c - -- + -*x3                                                                                       
 | \/        2    6                                                                                          
 |                                                                                                           
/
$$\int \frac{1}{\sqrt{\frac{a}{6} x_{3} + \left(c - \frac{x^{2}}{2}\right)}}\, dx = C + \sqrt{6} \left(\begin{cases} \frac{\sqrt{\frac{a x_{3}}{3} + 2 c} \operatorname{asin}{\left(\sqrt{3} x \sqrt{\frac{1}{a x_{3} + 6 c}} \right)}}{\sqrt{a x_{3} + 6 c}} & \text{for}\: a x_{3} + 6 c > 0 \end{cases}\right)$$
Simplificar
Respuesta [src] 
  1                                                                                          
  /                                                                                          
 |                                                                                           
 |  /                                                                           2            
 |  |                              -I                                          x             
 |  |---------------------------------------------------------------  for ------------ > 1   
 |  |       _____________________________                                   |    a*x3|       
 |  |      /                 2                ______________________      2*|c + ----|       
 |  |     /                 x                /           /    a*x3\         |     6  |       
 |  |    /   -1 + ---------------------- *  /  polar_lift|c + ----|                          
 |  |   /                     /    a*x3\  \/             \     6  /                          
 |  |  /          2*polar_lift|c + ----|                                                     
 |  |\/                       \     6  /                                                     
 |  <                                                                                      dx
 |  |                              1                                                         
 |  |--------------------------------------------------------------        otherwise         
 |  |       ____________________________                                                     
 |  |      /                2                ______________________                          
 |  |     /                x                /           /    a*x3\                           
 |  |    /   1 - ---------------------- *  /  polar_lift|c + ----|                           
 |  |   /                    /    a*x3\  \/             \     6  /                           
 |  |  /         2*polar_lift|c + ----|                                                      
 |  |\/                      \     6  /                                                      
 |  \                                                                                        
 |                                                                                           
/                                                                                            
0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i}{\sqrt{\frac{x^{2}}{2 \operatorname{polar\_lift}{\left(\frac{a x_{3}}{6} + c \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(\frac{a x_{3}}{6} + c \right)}}} & \text{for}\: \frac{x^{2}}{2 \left|{\frac{a x_{3}}{6} + c}\right|} > 1 \\\frac{1}{\sqrt{- \frac{x^{2}}{2 \operatorname{polar\_lift}{\left(\frac{a x_{3}}{6} + c \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(\frac{a x_{3}}{6} + c \right)}}} & \text{otherwise} \end{cases}\, dx$$ 
=
= 
  1                                                                                          
  /                                                                                          
 |                                                                                           
 |  /                                                                           2            
 |  |                              -I                                          x             
 |  |---------------------------------------------------------------  for ------------ > 1   
 |  |       _____________________________                                   |    a*x3|       
 |  |      /                 2                ______________________      2*|c + ----|       
 |  |     /                 x                /           /    a*x3\         |     6  |       
 |  |    /   -1 + ---------------------- *  /  polar_lift|c + ----|                          
 |  |   /                     /    a*x3\  \/             \     6  /                          
 |  |  /          2*polar_lift|c + ----|                                                     
 |  |\/                       \     6  /                                                     
 |  <                                                                                      dx
 |  |                              1                                                         
 |  |--------------------------------------------------------------        otherwise         
 |  |       ____________________________                                                     
 |  |      /                2                ______________________                          
 |  |     /                x                /           /    a*x3\                           
 |  |    /   1 - ---------------------- *  /  polar_lift|c + ----|                           
 |  |   /                    /    a*x3\  \/             \     6  /                           
 |  |  /         2*polar_lift|c + ----|                                                      
 |  |\/                      \     6  /                                                      
 |  \                                                                                        
 |                                                                                           
/                                                                                            
0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i}{\sqrt{\frac{x^{2}}{2 \operatorname{polar\_lift}{\left(\frac{a x_{3}}{6} + c \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(\frac{a x_{3}}{6} + c \right)}}} & \text{for}\: \frac{x^{2}}{2 \left|{\frac{a x_{3}}{6} + c}\right|} > 1 \\\frac{1}{\sqrt{- \frac{x^{2}}{2 \operatorname{polar\_lift}{\left(\frac{a x_{3}}{6} + c \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(\frac{a x_{3}}{6} + c \right)}}} & \text{otherwise} \end{cases}\, dx$$ 
Integral(Piecewise((-i/(sqrt(-1 + x^2/(2*polar_lift(c + a*x3/6)))*sqrt(polar_lift(c + a*x3/6))), x^2/(2*|c + a*x3/6|) > 1), (1/(sqrt(1 - x^2/(2*polar_lift(c + a*x3/6)))*sqrt(polar_lift(c + a*x3/6))), True)), (x, 0, 1))

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

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