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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                       
  /                       
 |                        
 |           2            
 |        cos (x)         
 |  ------------------- dx
 |  2*sin(x) + 3*cos(x)   
 |                        
/                         
0
∫01cos⁡2(x)2sin⁡(x)+3cos⁡(x) dx\int\limits_{0}^{1} \frac{\cos^{2}{\left(x \right)}}{2 \sin{\left(x \right)} + 3 \cos{\left(x \right)}}\, dx 
Integral(cos(x)^2/(2*sin(x) + 3*cos(x)), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                                 /        ____         \               /        ____         \                       /        ____         \                       /        ____         \
 |                                                            /x\           ____    |  2   \/ 13       /x\|       ____    |  2   \/ 13       /x\|       ____    2/x\    |  2   \/ 13       /x\|       ____    2/x\    |  2   \/ 13       /x\|
 |          2                                           78*tan|-|       4*\/ 13 *log|- - - ------ + tan|-||   4*\/ 13 *log|- - + ------ + tan|-||   4*\/ 13 *tan |-|*log|- - - ------ + tan|-||   4*\/ 13 *tan |-|*log|- - + ------ + tan|-||
 |       cos (x)                        52                    \2/                   \  3     3         \2//               \  3     3         \2//                \2/    \  3     3         \2//                \2/    \  3     3         \2//
 | ------------------- dx = C + ----------------- + ----------------- - ----------------------------------- + ----------------------------------- - ------------------------------------------- + -------------------------------------------
 | 2*sin(x) + 3*cos(x)                       2/x\                2/x\                         2/x\                                  2/x\                                      2/x\                                          2/x\             
 |                              169 + 169*tan |-|   169 + 169*tan |-|            169 + 169*tan |-|                     169 + 169*tan |-|                         169 + 169*tan |-|                             169 + 169*tan |-|             
/                                             \2/                 \2/                          \2/                                   \2/                                       \2/                                           \2/
∫cos⁡2(x)2sin⁡(x)+3cos⁡(x) dx=C+413log⁡(tan⁡(x2)−23+133)tan⁡2(x2)169tan⁡2(x2)+169+413log⁡(tan⁡(x2)−23+133)169tan⁡2(x2)+169−413log⁡(tan⁡(x2)−133−23)tan⁡2(x2)169tan⁡2(x2)+169−413log⁡(tan⁡(x2)−133−23)169tan⁡2(x2)+169+78tan⁡(x2)169tan⁡2(x2)+169+52169tan⁡2(x2)+169\int \frac{\cos^{2}{\left(x \right)}}{2 \sin{\left(x \right)} + 3 \cos{\left(x \right)}}\, dx = C + \frac{4 \sqrt{13} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{2}{3} + \frac{\sqrt{13}}{3} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{169 \tan^{2}{\left(\frac{x}{2} \right)} + 169} + \frac{4 \sqrt{13} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{2}{3} + \frac{\sqrt{13}}{3} \right)}}{169 \tan^{2}{\left(\frac{x}{2} \right)} + 169} - \frac{4 \sqrt{13} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{13}}{3} - \frac{2}{3} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{169 \tan^{2}{\left(\frac{x}{2} \right)} + 169} - \frac{4 \sqrt{13} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{13}}{3} - \frac{2}{3} \right)}}{169 \tan^{2}{\left(\frac{x}{2} \right)} + 169} + \frac{78 \tan{\left(\frac{x}{2} \right)}}{169 \tan^{2}{\left(\frac{x}{2} \right)} + 169} + \frac{52}{169 \tan^{2}{\left(\frac{x}{2} \right)} + 169}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.900.000.50
Trazar el gráfico f(x)
Respuesta [src] 
                                                               /        ____\            /          /      ____\\            /          /                 ____\\               /        ____           \                      /          /                 ____\\                         /        ____           \
                                                       ____    |  2   \/ 13 |       ____ |          |2   \/ 13 ||       ____ |          |2              \/ 13 ||       ____    |  2   \/ 13            |       ____    2      |          |2              \/ 13 ||       ____    2         |  2   \/ 13            |
                                                   4*\/ 13 *log|- - + ------|   4*\/ 13 *|pi*I + log|- + ------||   4*\/ 13 *|pi*I + log|- - tan(1/2) + ------||   4*\/ 13 *log|- - + ------ + tan(1/2)|   4*\/ 13 *tan (1/2)*|pi*I + log|- - tan(1/2) + ------||   4*\/ 13 *tan (1/2)*log|- - + ------ + tan(1/2)|
  4             52               78*tan(1/2)                   \  3     3   /            \          \3     3   //            \          \3                3   //               \  3     3              /                      \          \3                3   //                         \  3     3              /
- -- + ------------------- + ------------------- - -------------------------- + --------------------------------- - -------------------------------------------- + ------------------------------------- - ------------------------------------------------------ + -----------------------------------------------
  13                2                     2                   169                              169                                           2                                           2                                               2                                                     2                   
       169 + 169*tan (1/2)   169 + 169*tan (1/2)                                                                                169 + 169*tan (1/2)                         169 + 169*tan (1/2)                             169 + 169*tan (1/2)                                   169 + 169*tan (1/2)
−413+413log⁡(−23+tan⁡(12)+133)tan⁡2(12)169tan⁡2(12)+169+413log⁡(−23+tan⁡(12)+133)169tan⁡2(12)+169−413log⁡(−23+133)169+78tan⁡(12)169tan⁡2(12)+169+52169tan⁡2(12)+169−413(log⁡(−tan⁡(12)+23+133)+iπ)169tan⁡2(12)+169−413(log⁡(−tan⁡(12)+23+133)+iπ)tan⁡2(12)169tan⁡2(12)+169+413(log⁡(23+133)+iπ)169- \frac{4}{13} + \frac{4 \sqrt{13} \log{\left(- \frac{2}{3} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{13}}{3} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} + \frac{4 \sqrt{13} \log{\left(- \frac{2}{3} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{13}}{3} \right)}}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} - \frac{4 \sqrt{13} \log{\left(- \frac{2}{3} + \frac{\sqrt{13}}{3} \right)}}{169} + \frac{78 \tan{\left(\frac{1}{2} \right)}}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} + \frac{52}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} - \frac{4 \sqrt{13} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{2}{3} + \frac{\sqrt{13}}{3} \right)} + i \pi\right)}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} - \frac{4 \sqrt{13} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{2}{3} + \frac{\sqrt{13}}{3} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} + \frac{4 \sqrt{13} \left(\log{\left(\frac{2}{3} + \frac{\sqrt{13}}{3} \right)} + i \pi\right)}{169} 
=
= 
                                                               /        ____\            /          /      ____\\            /          /                 ____\\               /        ____           \                      /          /                 ____\\                         /        ____           \
                                                       ____    |  2   \/ 13 |       ____ |          |2   \/ 13 ||       ____ |          |2              \/ 13 ||       ____    |  2   \/ 13            |       ____    2      |          |2              \/ 13 ||       ____    2         |  2   \/ 13            |
                                                   4*\/ 13 *log|- - + ------|   4*\/ 13 *|pi*I + log|- + ------||   4*\/ 13 *|pi*I + log|- - tan(1/2) + ------||   4*\/ 13 *log|- - + ------ + tan(1/2)|   4*\/ 13 *tan (1/2)*|pi*I + log|- - tan(1/2) + ------||   4*\/ 13 *tan (1/2)*log|- - + ------ + tan(1/2)|
  4             52               78*tan(1/2)                   \  3     3   /            \          \3     3   //            \          \3                3   //               \  3     3              /                      \          \3                3   //                         \  3     3              /
- -- + ------------------- + ------------------- - -------------------------- + --------------------------------- - -------------------------------------------- + ------------------------------------- - ------------------------------------------------------ + -----------------------------------------------
  13                2                     2                   169                              169                                           2                                           2                                               2                                                     2                   
       169 + 169*tan (1/2)   169 + 169*tan (1/2)                                                                                169 + 169*tan (1/2)                         169 + 169*tan (1/2)                             169 + 169*tan (1/2)                                   169 + 169*tan (1/2)
−413+413log⁡(−23+tan⁡(12)+133)tan⁡2(12)169tan⁡2(12)+169+413log⁡(−23+tan⁡(12)+133)169tan⁡2(12)+169−413log⁡(−23+133)169+78tan⁡(12)169tan⁡2(12)+169+52169tan⁡2(12)+169−413(log⁡(−tan⁡(12)+23+133)+iπ)169tan⁡2(12)+169−413(log⁡(−tan⁡(12)+23+133)+iπ)tan⁡2(12)169tan⁡2(12)+169+413(log⁡(23+133)+iπ)169- \frac{4}{13} + \frac{4 \sqrt{13} \log{\left(- \frac{2}{3} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{13}}{3} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} + \frac{4 \sqrt{13} \log{\left(- \frac{2}{3} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{13}}{3} \right)}}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} - \frac{4 \sqrt{13} \log{\left(- \frac{2}{3} + \frac{\sqrt{13}}{3} \right)}}{169} + \frac{78 \tan{\left(\frac{1}{2} \right)}}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} + \frac{52}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} - \frac{4 \sqrt{13} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{2}{3} + \frac{\sqrt{13}}{3} \right)} + i \pi\right)}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} - \frac{4 \sqrt{13} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{2}{3} + \frac{\sqrt{13}}{3} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} + \frac{4 \sqrt{13} \left(\log{\left(\frac{2}{3} + \frac{\sqrt{13}}{3} \right)} + i \pi\right)}{169} 
-4/13 + 52/(169 + 169*tan(1/2)^2) + 78*tan(1/2)/(169 + 169*tan(1/2)^2) - 4*sqrt(13)*log(-2/3 + sqrt(13)/3)/169 + 4*sqrt(13)*(pi*i + log(2/3 + sqrt(13)/3))/169 - 4*sqrt(13)*(pi*i + log(2/3 - tan(1/2) + sqrt(13)/3))/(169 + 169*tan(1/2)^2) + 4*sqrt(13)*log(-2/3 + sqrt(13)/3 + tan(1/2))/(169 + 169*tan(1/2)^2) - 4*sqrt(13)*tan(1/2)^2*(pi*i + log(2/3 - tan(1/2) + sqrt(13)/3))/(169 + 169*tan(1/2)^2) + 4*sqrt(13)*tan(1/2)^2*log(-2/3 + sqrt(13)/3 + tan(1/2))/(169 + 169*tan(1/2)^2)
Respuesta numérica [src] 
0.213010141513137
0.213010141513137

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

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