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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                
  /                
 |                 
 |       2         
 |    sin (x)      
 |  ------------ dx
 |  2 + 3*cos(x)   
 |                 
/                  
0
∫01sin⁡2(x)3cos⁡(x)+2 dx\int\limits_{0}^{1} \frac{\sin^{2}{\left(x \right)}}{3 \cos{\left(x \right)} + 2}\, dx 
Integral(sin(x)^2/(2 + 3*cos(x)), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                                                                                                                                                       
 |                               /x\                       ___    /  ___      /x\\     ___    /    ___      /x\\           2/x\      ___    2/x\    /  ___      /x\\     ___    2/x\    /    ___      /x\\
 |      2                   6*tan|-|                     \/ 5 *log|\/ 5  + tan|-||   \/ 5 *log|- \/ 5  + tan|-||    2*x*tan |-|    \/ 5 *tan |-|*log|\/ 5  + tan|-||   \/ 5 *tan |-|*log|- \/ 5  + tan|-||
 |   sin (x)                     \2/          2*x                 \           \2//            \             \2//            \2/              \2/    \           \2//             \2/    \             \2//
 | ------------ dx = C - ------------- + ------------- + ------------------------- - --------------------------- + ------------- + --------------------------------- - -----------------------------------
 | 2 + 3*cos(x)                   2/x\            2/x\                  2/x\                         2/x\                   2/x\                      2/x\                                 2/x\           
 |                       9 + 9*tan |-|   9 + 9*tan |-|         9 + 9*tan |-|                9 + 9*tan |-|          9 + 9*tan |-|             9 + 9*tan |-|                        9 + 9*tan |-|           
/                                  \2/             \2/                   \2/                          \2/                    \2/                       \2/                                  \2/
∫sin⁡2(x)3cos⁡(x)+2 dx=C+2xtan⁡2(x2)9tan⁡2(x2)+9+2x9tan⁡2(x2)+9−5log⁡(tan⁡(x2)−5)tan⁡2(x2)9tan⁡2(x2)+9−5log⁡(tan⁡(x2)−5)9tan⁡2(x2)+9+5log⁡(tan⁡(x2)+5)tan⁡2(x2)9tan⁡2(x2)+9+5log⁡(tan⁡(x2)+5)9tan⁡2(x2)+9−6tan⁡(x2)9tan⁡2(x2)+9\int \frac{\sin^{2}{\left(x \right)}}{3 \cos{\left(x \right)} + 2}\, dx = C + \frac{2 x \tan^{2}{\left(\frac{x}{2} \right)}}{9 \tan^{2}{\left(\frac{x}{2} \right)} + 9} + \frac{2 x}{9 \tan^{2}{\left(\frac{x}{2} \right)} + 9} - \frac{\sqrt{5} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{5} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{9 \tan^{2}{\left(\frac{x}{2} \right)} + 9} - \frac{\sqrt{5} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{5} \right)}}{9 \tan^{2}{\left(\frac{x}{2} \right)} + 9} + \frac{\sqrt{5} \log{\left(\tan{\left(\frac{x}{2} \right)} + \sqrt{5} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{9 \tan^{2}{\left(\frac{x}{2} \right)} + 9} + \frac{\sqrt{5} \log{\left(\tan{\left(\frac{x}{2} \right)} + \sqrt{5} \right)}}{9 \tan^{2}{\left(\frac{x}{2} \right)} + 9} - \frac{6 \tan{\left(\frac{x}{2} \right)}}{9 \tan^{2}{\left(\frac{x}{2} \right)} + 9}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.900.00.2
Trazar el gráfico f(x)
Respuesta [src] 
                                           2            ___    /  ___\     ___ /          /  ___\\     ___    /  ___           \     ___ /          /  ___           \\     ___    2         /  ___           \     ___    2      /          /  ___           \\
       2             6*tan(1/2)       2*tan (1/2)     \/ 5 *log\\/ 5 /   \/ 5 *\pi*I + log\\/ 5 //   \/ 5 *log\\/ 5  + tan(1/2)/   \/ 5 *\pi*I + log\\/ 5  - tan(1/2)//   \/ 5 *tan (1/2)*log\\/ 5  + tan(1/2)/   \/ 5 *tan (1/2)*\pi*I + log\\/ 5  - tan(1/2)//
--------------- - --------------- + --------------- - ---------------- + ------------------------- + --------------------------- - ------------------------------------ + ------------------------------------- - ----------------------------------------------
         2                 2                 2               9                       9                              2                                 2                                       2                                           2                     
9 + 9*tan (1/2)   9 + 9*tan (1/2)   9 + 9*tan (1/2)                                                        9 + 9*tan (1/2)                   9 + 9*tan (1/2)                         9 + 9*tan (1/2)                             9 + 9*tan (1/2)
−6tan⁡(12)9tan⁡2(12)+9−5log⁡(5)9+2tan⁡2(12)9tan⁡2(12)+9+5log⁡(tan⁡(12)+5)tan⁡2(12)9tan⁡2(12)+9+29tan⁡2(12)+9+5log⁡(tan⁡(12)+5)9tan⁡2(12)+9−5(log⁡(−tan⁡(12)+5)+iπ)9tan⁡2(12)+9−5(log⁡(−tan⁡(12)+5)+iπ)tan⁡2(12)9tan⁡2(12)+9+5(log⁡(5)+iπ)9- \frac{6 \tan{\left(\frac{1}{2} \right)}}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 9} - \frac{\sqrt{5} \log{\left(\sqrt{5} \right)}}{9} + \frac{2 \tan^{2}{\left(\frac{1}{2} \right)}}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 9} + \frac{\sqrt{5} \log{\left(\tan{\left(\frac{1}{2} \right)} + \sqrt{5} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 9} + \frac{2}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 9} + \frac{\sqrt{5} \log{\left(\tan{\left(\frac{1}{2} \right)} + \sqrt{5} \right)}}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 9} - \frac{\sqrt{5} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \sqrt{5} \right)} + i \pi\right)}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 9} - \frac{\sqrt{5} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \sqrt{5} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 9} + \frac{\sqrt{5} \left(\log{\left(\sqrt{5} \right)} + i \pi\right)}{9} 
=
= 
                                           2            ___    /  ___\     ___ /          /  ___\\     ___    /  ___           \     ___ /          /  ___           \\     ___    2         /  ___           \     ___    2      /          /  ___           \\
       2             6*tan(1/2)       2*tan (1/2)     \/ 5 *log\\/ 5 /   \/ 5 *\pi*I + log\\/ 5 //   \/ 5 *log\\/ 5  + tan(1/2)/   \/ 5 *\pi*I + log\\/ 5  - tan(1/2)//   \/ 5 *tan (1/2)*log\\/ 5  + tan(1/2)/   \/ 5 *tan (1/2)*\pi*I + log\\/ 5  - tan(1/2)//
--------------- - --------------- + --------------- - ---------------- + ------------------------- + --------------------------- - ------------------------------------ + ------------------------------------- - ----------------------------------------------
         2                 2                 2               9                       9                              2                                 2                                       2                                           2                     
9 + 9*tan (1/2)   9 + 9*tan (1/2)   9 + 9*tan (1/2)                                                        9 + 9*tan (1/2)                   9 + 9*tan (1/2)                         9 + 9*tan (1/2)                             9 + 9*tan (1/2)
−6tan⁡(12)9tan⁡2(12)+9−5log⁡(5)9+2tan⁡2(12)9tan⁡2(12)+9+5log⁡(tan⁡(12)+5)tan⁡2(12)9tan⁡2(12)+9+29tan⁡2(12)+9+5log⁡(tan⁡(12)+5)9tan⁡2(12)+9−5(log⁡(−tan⁡(12)+5)+iπ)9tan⁡2(12)+9−5(log⁡(−tan⁡(12)+5)+iπ)tan⁡2(12)9tan⁡2(12)+9+5(log⁡(5)+iπ)9- \frac{6 \tan{\left(\frac{1}{2} \right)}}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 9} - \frac{\sqrt{5} \log{\left(\sqrt{5} \right)}}{9} + \frac{2 \tan^{2}{\left(\frac{1}{2} \right)}}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 9} + \frac{\sqrt{5} \log{\left(\tan{\left(\frac{1}{2} \right)} + \sqrt{5} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 9} + \frac{2}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 9} + \frac{\sqrt{5} \log{\left(\tan{\left(\frac{1}{2} \right)} + \sqrt{5} \right)}}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 9} - \frac{\sqrt{5} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \sqrt{5} \right)} + i \pi\right)}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 9} - \frac{\sqrt{5} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \sqrt{5} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{9 \tan^{2}{\left(\frac{1}{2} \right)} + 9} + \frac{\sqrt{5} \left(\log{\left(\sqrt{5} \right)} + i \pi\right)}{9} 
2/(9 + 9*tan(1/2)^2) - 6*tan(1/2)/(9 + 9*tan(1/2)^2) + 2*tan(1/2)^2/(9 + 9*tan(1/2)^2) - sqrt(5)*log(sqrt(5))/9 + sqrt(5)*(pi*i + log(sqrt(5)))/9 + sqrt(5)*log(sqrt(5) + tan(1/2))/(9 + 9*tan(1/2)^2) - sqrt(5)*(pi*i + log(sqrt(5) - tan(1/2)))/(9 + 9*tan(1/2)^2) + sqrt(5)*tan(1/2)^2*log(sqrt(5) + tan(1/2))/(9 + 9*tan(1/2)^2) - sqrt(5)*tan(1/2)^2*(pi*i + log(sqrt(5) - tan(1/2)))/(9 + 9*tan(1/2)^2)
Respuesta numérica [src] 
0.0656382581798884
0.0656382581798884

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

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