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Resolución de integrales/ (2cosx+4sinx)/(2sinx-4cosx)^3

Integral de (2cosx+4sinx)/(2sinx-4cosx)^3 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                          
  /                          
 |                           
 |   2*cos(x) + 4*sin(x)     
 |  ---------------------- dx
 |                       3   
 |  (2*sin(x) - 4*cos(x))    
 |                           
/                            
0
$$\int\limits_{0}^{1} \frac{4 \sin{\left(x \right)} + 2 \cos{\left(x \right)}}{\left(2 \sin{\left(x \right)} - 4 \cos{\left(x \right)}\right)^{3}}\, dx$$ 
Integral((2*cos(x) + 4*sin(x))/(2*sin(x) - 4*cos(x))^3, (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                          2/x\                                                     2/x\                                                     /x\                                                    3/x\                      
 |                                                        tan |-|                                                2*tan |-|                                                2*tan|-|                                               2*tan |-|                      
 |  2*cos(x) + 4*sin(x)                                       \2/                                                      \2/                                                     \2/                                                     \2/                      
 | ---------------------- dx = C + ----------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------------- + -----------------------------------------------------
 |                      3                     /x\         2/x\         4/x\         3/x\              /x\         2/x\         4/x\         3/x\              /x\         2/x\         4/x\         3/x\              /x\         2/x\         4/x\         3/x\
 | (2*sin(x) - 4*cos(x))           32 - 64*tan|-| - 32*tan |-| + 32*tan |-| + 64*tan |-|   16 - 32*tan|-| - 16*tan |-| + 16*tan |-| + 32*tan |-|   32 - 64*tan|-| - 32*tan |-| + 32*tan |-| + 64*tan |-|   32 - 64*tan|-| - 32*tan |-| + 32*tan |-| + 64*tan |-|
 |                                            \2/          \2/          \2/          \2/              \2/          \2/          \2/          \2/              \2/          \2/          \2/          \2/              \2/          \2/          \2/          \2/
/
$$\int \frac{4 \sin{\left(x \right)} + 2 \cos{\left(x \right)}}{\left(2 \sin{\left(x \right)} - 4 \cos{\left(x \right)}\right)^{3}}\, dx = C + \frac{2 \tan^{3}{\left(\frac{x}{2} \right)}}{32 \tan^{4}{\left(\frac{x}{2} \right)} + 64 \tan^{3}{\left(\frac{x}{2} \right)} - 32 \tan^{2}{\left(\frac{x}{2} \right)} - 64 \tan{\left(\frac{x}{2} \right)} + 32} + \frac{\tan^{2}{\left(\frac{x}{2} \right)}}{32 \tan^{4}{\left(\frac{x}{2} \right)} + 64 \tan^{3}{\left(\frac{x}{2} \right)} - 32 \tan^{2}{\left(\frac{x}{2} \right)} - 64 \tan{\left(\frac{x}{2} \right)} + 32} - \frac{2 \tan{\left(\frac{x}{2} \right)}}{32 \tan^{4}{\left(\frac{x}{2} \right)} + 64 \tan^{3}{\left(\frac{x}{2} \right)} - 32 \tan^{2}{\left(\frac{x}{2} \right)} - 64 \tan{\left(\frac{x}{2} \right)} + 32} - \frac{2 \tan^{2}{\left(\frac{x}{2} \right)}}{16 \tan^{4}{\left(\frac{x}{2} \right)} + 32 \tan^{3}{\left(\frac{x}{2} \right)} - 16 \tan^{2}{\left(\frac{x}{2} \right)} - 32 \tan{\left(\frac{x}{2} \right)} + 16}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
1                        1                    
-- - -----------------------------------------
32        2            2                      
     8*sin (1) + 32*cos (1) - 32*cos(1)*sin(1)
$$\frac{1}{32} - \frac{1}{- 32 \sin{\left(1 \right)} \cos{\left(1 \right)} + 8 \sin^{2}{\left(1 \right)} + 32 \cos^{2}{\left(1 \right)}}$$ 
=
= 
1                        1                    
-- - -----------------------------------------
32        2            2                      
     8*sin (1) + 32*cos (1) - 32*cos(1)*sin(1)
$$\frac{1}{32} - \frac{1}{- 32 \sin{\left(1 \right)} \cos{\left(1 \right)} + 8 \sin^{2}{\left(1 \right)} + 32 \cos^{2}{\left(1 \right)}}$$ 
1/32 - 1/(8*sin(1)^2 + 32*cos(1)^2 - 32*cos(1)*sin(1))
Respuesta numérica [src] 
-2.15464205400457
-2.15464205400457

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

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