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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  p                          
  -                          
  4                          
  /                          
 |                           
 |   2*cos(x) - 3*sin(x)     
 |  ---------------------- dx
 |                       3   
 |  (2*sin(x) + 3*cos(x))    
 |                           
/                            
0
∫0p4−3sin⁡(x)+2cos⁡(x)(2sin⁡(x)+3cos⁡(x))3 dx\int\limits_{0}^{\frac{p}{4}} \frac{- 3 \sin{\left(x \right)} + 2 \cos{\left(x \right)}}{\left(2 \sin{\left(x \right)} + 3 \cos{\left(x \right)}\right)^{3}}\, dx 
Integral((2*cos(x) - 3*sin(x))/(2*sin(x) + 3*cos(x))^3, (x, 0, p/4))
Respuesta (Indefinida) [src] 
  /                                                             3/x\                                                   2/x\                                                     2/x\                                                       /x\                       
 |                                                        12*tan |-|                                              6*tan |-|                                                8*tan |-|                                                 12*tan|-|                       
 |  2*cos(x) - 3*sin(x)                                          \2/                                                    \2/                                                      \2/                                                       \2/                       
 | ---------------------- dx = C - ------------------------------------------------------- - ---------------------------------------------------- + ------------------------------------------------------- + -------------------------------------------------------
 |                      3                      3/x\         2/x\         4/x\          /x\              3/x\        2/x\         4/x\         /x\               3/x\         2/x\         4/x\          /x\               3/x\         2/x\         4/x\          /x\
 | (2*sin(x) + 3*cos(x))           81 - 216*tan |-| - 18*tan |-| + 81*tan |-| + 216*tan|-|   27 - 72*tan |-| - 6*tan |-| + 27*tan |-| + 72*tan|-|   81 - 216*tan |-| - 18*tan |-| + 81*tan |-| + 216*tan|-|   81 - 216*tan |-| - 18*tan |-| + 81*tan |-| + 216*tan|-|
 |                                              \2/          \2/          \2/          \2/               \2/         \2/          \2/         \2/                \2/          \2/          \2/          \2/                \2/          \2/          \2/          \2/
/
∫−3sin⁡(x)+2cos⁡(x)(2sin⁡(x)+3cos⁡(x))3 dx=C−12tan⁡3(x2)81tan⁡4(x2)−216tan⁡3(x2)−18tan⁡2(x2)+216tan⁡(x2)+81+8tan⁡2(x2)81tan⁡4(x2)−216tan⁡3(x2)−18tan⁡2(x2)+216tan⁡(x2)+81+12tan⁡(x2)81tan⁡4(x2)−216tan⁡3(x2)−18tan⁡2(x2)+216tan⁡(x2)+81−6tan⁡2(x2)27tan⁡4(x2)−72tan⁡3(x2)−6tan⁡2(x2)+72tan⁡(x2)+27\int \frac{- 3 \sin{\left(x \right)} + 2 \cos{\left(x \right)}}{\left(2 \sin{\left(x \right)} + 3 \cos{\left(x \right)}\right)^{3}}\, dx = C - \frac{12 \tan^{3}{\left(\frac{x}{2} \right)}}{81 \tan^{4}{\left(\frac{x}{2} \right)} - 216 \tan^{3}{\left(\frac{x}{2} \right)} - 18 \tan^{2}{\left(\frac{x}{2} \right)} + 216 \tan{\left(\frac{x}{2} \right)} + 81} + \frac{8 \tan^{2}{\left(\frac{x}{2} \right)}}{81 \tan^{4}{\left(\frac{x}{2} \right)} - 216 \tan^{3}{\left(\frac{x}{2} \right)} - 18 \tan^{2}{\left(\frac{x}{2} \right)} + 216 \tan{\left(\frac{x}{2} \right)} + 81} + \frac{12 \tan{\left(\frac{x}{2} \right)}}{81 \tan^{4}{\left(\frac{x}{2} \right)} - 216 \tan^{3}{\left(\frac{x}{2} \right)} - 18 \tan^{2}{\left(\frac{x}{2} \right)} + 216 \tan{\left(\frac{x}{2} \right)} + 81} - \frac{6 \tan^{2}{\left(\frac{x}{2} \right)}}{27 \tan^{4}{\left(\frac{x}{2} \right)} - 72 \tan^{3}{\left(\frac{x}{2} \right)} - 6 \tan^{2}{\left(\frac{x}{2} \right)} + 72 \tan{\left(\frac{x}{2} \right)} + 27}
Simplificar
Respuesta [src] 
1                        1                    
-- - -----------------------------------------
18        2/p\         2/p\         /p\    /p\
     8*sin |-| + 18*cos |-| + 24*cos|-|*sin|-|
           \4/          \4/         \4/    \4/
118−18sin⁡2(p4)+24sin⁡(p4)cos⁡(p4)+18cos⁡2(p4)\frac{1}{18} - \frac{1}{8 \sin^{2}{\left(\frac{p}{4} \right)} + 24 \sin{\left(\frac{p}{4} \right)} \cos{\left(\frac{p}{4} \right)} + 18 \cos^{2}{\left(\frac{p}{4} \right)}} 
=
= 
1                        1                    
-- - -----------------------------------------
18        2/p\         2/p\         /p\    /p\
     8*sin |-| + 18*cos |-| + 24*cos|-|*sin|-|
           \4/          \4/         \4/    \4/
118−18sin⁡2(p4)+24sin⁡(p4)cos⁡(p4)+18cos⁡2(p4)\frac{1}{18} - \frac{1}{8 \sin^{2}{\left(\frac{p}{4} \right)} + 24 \sin{\left(\frac{p}{4} \right)} \cos{\left(\frac{p}{4} \right)} + 18 \cos^{2}{\left(\frac{p}{4} \right)}} 
1/18 - 1/(8*sin(p/4)^2 + 18*cos(p/4)^2 + 24*cos(p/4)*sin(p/4))

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

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