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Resolución de integrales/ 4-x^2/ cos(y)/ x^2*cos/ 2*x/(sqrt(4-x^2*cos(y)^2))

Integral de 2*x/(sqrt(4-x^2*cos(y)^2)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  2                       
  /                       
 |                        
 |          2*x           
 |  ------------------- dx
 |     ________________   
 |    /      2    2       
 |  \/  4 - x *cos (y)    
 |                        
/                         
0
∫022x−x2cos⁡2(y)+4 dx\int\limits_{0}^{2} \frac{2 x}{\sqrt{- x^{2} \cos^{2}{\left(y \right)} + 4}}\, dx 
Integral((2*x)/sqrt(4 - x^2*cos(y)^2), (x, 0, 2))
Respuesta (Indefinida) [src] 
                                //      ________________                  \
                                ||     /      2    2                      |
  /                             ||-2*\/  4 - x *cos (y)          2        |
 |                              ||----------------------  for cos (y) != 0|
 |         2*x                  ||          2                             |
 | ------------------- dx = C + |<       cos (y)                          |
 |    ________________          ||                                        |
 |   /      2    2              ||           2                            |
 | \/  4 - x *cos (y)           ||          x                             |
 |                              ||          --               otherwise    |
/                               \\          2                             /
∫2x−x2cos⁡2(y)+4 dx=C+{−2−x2cos⁡2(y)+4cos⁡2(y)for cos⁡2(y)≠0x22otherwise\int \frac{2 x}{\sqrt{- x^{2} \cos^{2}{\left(y \right)} + 4}}\, dx = C + \begin{cases} - \frac{2 \sqrt{- x^{2} \cos^{2}{\left(y \right)} + 4}}{\cos^{2}{\left(y \right)}} & \text{for}\: \cos^{2}{\left(y \right)} \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/                                      /    pi      3*pi\
|              2                 for Or|y = --, y = ----|
|                                      \    2        2  /
|                                                        
|               _______________                          
<              /          2                              
|   4      2*\/  4 - 4*cos (y)                           
|------- - --------------------         otherwise        
|   2               2                                    
|cos (y)         cos (y)                                 
\
{2for y=π2∨y=3π2−24−4cos⁡2(y)cos⁡2(y)+4cos⁡2(y)otherwise\begin{cases} 2 & \text{for}\: y = \frac{\pi}{2} \vee y = \frac{3 \pi}{2} \\- \frac{2 \sqrt{4 - 4 \cos^{2}{\left(y \right)}}}{\cos^{2}{\left(y \right)}} + \frac{4}{\cos^{2}{\left(y \right)}} & \text{otherwise} \end{cases} 
=
= 
/                                      /    pi      3*pi\
|              2                 for Or|y = --, y = ----|
|                                      \    2        2  /
|                                                        
|               _______________                          
<              /          2                              
|   4      2*\/  4 - 4*cos (y)                           
|------- - --------------------         otherwise        
|   2               2                                    
|cos (y)         cos (y)                                 
\
{2for y=π2∨y=3π2−24−4cos⁡2(y)cos⁡2(y)+4cos⁡2(y)otherwise\begin{cases} 2 & \text{for}\: y = \frac{\pi}{2} \vee y = \frac{3 \pi}{2} \\- \frac{2 \sqrt{4 - 4 \cos^{2}{\left(y \right)}}}{\cos^{2}{\left(y \right)}} + \frac{4}{\cos^{2}{\left(y \right)}} & \text{otherwise} \end{cases} 
Piecewise((2, (y = pi/2)∨(y = 3*pi/2)), (4/cos(y)^2 - 2*sqrt(4 - 4*cos(y)^2)/cos(y)^2, True))

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

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