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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  9                        
  /                        
 |                         
 |       1 + cot(x)        
 |  -------------------- dx
 |                     2   
 |  (sin(x) + 2*cos(x))    
 |                         
/                          
4
∫49cot⁡(x)+1(sin⁡(x)+2cos⁡(x))2 dx\int\limits_{4}^{9} \frac{\cot{\left(x \right)} + 1}{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right)^{2}}\, dx 
Integral((1 + cot(x))/(sin(x) + 2*cos(x))^2, (x, 4, 9))
Respuesta (Indefinida) [src] 
  /                                           /x\              /                       
 |                                         tan|-|             |                        
 |      1 + cot(x)                            \2/             |        cot(x)          
 | -------------------- dx = C - ------------------------- +  | -------------------- dx
 |                    2                    /x\        2/x\    |                    2   
 | (sin(x) + 2*cos(x))           -2 - 2*tan|-| + 2*tan |-|    | (2*cos(x) + sin(x))    
 |                                         \2/         \2/    |                        
/                                                            /
∫cot⁡(x)+1(sin⁡(x)+2cos⁡(x))2 dx=C+∫cot⁡(x)(sin⁡(x)+2cos⁡(x))2 dx−tan⁡(x2)2tan⁡2(x2)−2tan⁡(x2)−2\int \frac{\cot{\left(x \right)} + 1}{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right)^{2}}\, dx = C + \int \frac{\cot{\left(x \right)}}{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right)^{2}}\, dx - \frac{\tan{\left(\frac{x}{2} \right)}}{2 \tan^{2}{\left(\frac{x}{2} \right)} - 2 \tan{\left(\frac{x}{2} \right)} - 2}
Simplificar
Respuesta [src] 
  9                        
  /                        
 |                         
 |       1 + cot(x)        
 |  -------------------- dx
 |                     2   
 |  (2*cos(x) + sin(x))    
 |                         
/                          
4
∫49cot⁡(x)+1(sin⁡(x)+2cos⁡(x))2 dx\int\limits_{4}^{9} \frac{\cot{\left(x \right)} + 1}{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right)^{2}}\, dx 
=
= 
  9                        
  /                        
 |                         
 |       1 + cot(x)        
 |  -------------------- dx
 |                     2   
 |  (2*cos(x) + sin(x))    
 |                         
/                          
4
∫49cot⁡(x)+1(sin⁡(x)+2cos⁡(x))2 dx\int\limits_{4}^{9} \frac{\cot{\left(x \right)} + 1}{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right)^{2}}\, dx 
Integral((1 + cot(x))/(2*cos(x) + sin(x))^2, (x, 4, 9))
Respuesta numérica [src] 
65.5827624263205
65.5827624263205

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

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