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Resolución de integrales/ 2sinx/ 2ctgx/ x+cosx/ (2ctgx+1)/(2sinx+cosx)^2

Integral de (2ctgx+1)/(2sinx+cosx)^2 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                        
  /                        
 |                         
 |      2*cot(x) + 1       
 |  -------------------- dx
 |                     2   
 |  (2*sin(x) + cos(x))    
 |                         
/                          
0
$$\int\limits_{0}^{1} \frac{2 \cot{\left(x \right)} + 1}{\left(2 \sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}\, dx$$ 
Integral((2*cot(x) + 1)/(2*sin(x) + cos(x))^2, (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                  /                                       /x\       
 |                                  |                                   2*tan|-|       
 |     2*cot(x) + 1                 |        cot(x)                          \2/       
 | -------------------- dx = C + 2* | -------------------- dx - -----------------------
 |                    2             |                    2              2/x\        /x\
 | (2*sin(x) + cos(x))              | (2*sin(x) + cos(x))       -1 + tan |-| - 4*tan|-|
 |                                  |                                    \2/        \2/
/                                  /
$$\int \frac{2 \cot{\left(x \right)} + 1}{\left(2 \sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}\, dx = C + 2 \int \frac{\cot{\left(x \right)}}{\left(2 \sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}\, dx - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 4 \tan{\left(\frac{x}{2} \right)} - 1}$$
Simplificar
Respuesta [src] 
  1                        
  /                        
 |                         
 |      1 + 2*cot(x)       
 |  -------------------- dx
 |                     2   
 |  (2*sin(x) + cos(x))    
 |                         
/                          
0
$$\int\limits_{0}^{1} \frac{2 \cot{\left(x \right)} + 1}{\left(2 \sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}\, dx$$ 
=
= 
  1                        
  /                        
 |                         
 |      1 + 2*cot(x)       
 |  -------------------- dx
 |                     2   
 |  (2*sin(x) + cos(x))    
 |                         
/                          
0
$$\int\limits_{0}^{1} \frac{2 \cot{\left(x \right)} + 1}{\left(2 \sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}\, dx$$ 
Integral((1 + 2*cot(x))/(2*sin(x) + cos(x))^2, (x, 0, 1))
Respuesta numérica [src] 
85.1022861439764
85.1022861439764

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

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