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Resolución de integrales/ sin(x)/ (1+sin(x))*1/(1-sin(x))^2

Integral de (1+sin(x))*1/(1-sin(x))^2 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 pi                 
 --                 
 4                  
  /                 
 |                  
 |    1 + sin(x)    
 |  ------------- dx
 |              2   
 |  (1 - sin(x))    
 |                  
/                   
0
∫0π4sin⁡(x)+1(1−sin⁡(x))2 dx\int\limits_{0}^{\frac{\pi}{4}} \frac{\sin{\left(x \right)} + 1}{\left(1 - \sin{\left(x \right)}\right)^{2}}\, dx 
Integral((1 + sin(x))/(1 - sin(x))^2, (x, 0, pi/4))
Respuesta (Indefinida) [src] 
  /                                                                                  2/x\              
 |                                                                              6*tan |-|              
 |   1 + sin(x)                             2                                         \2/              
 | ------------- dx = C - ------------------------------------- - -------------------------------------
 |             2                    2/x\        3/x\        /x\             2/x\        3/x\        /x\
 | (1 - sin(x))           -3 - 9*tan |-| + 3*tan |-| + 9*tan|-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan|-|
 |                                   \2/         \2/        \2/              \2/         \2/        \2/
/
∫sin⁡(x)+1(1−sin⁡(x))2 dx=C−6tan⁡2(x2)3tan⁡3(x2)−9tan⁡2(x2)+9tan⁡(x2)−3−23tan⁡3(x2)−9tan⁡2(x2)+9tan⁡(x2)−3\int \frac{\sin{\left(x \right)} + 1}{\left(1 - \sin{\left(x \right)}\right)^{2}}\, dx = C - \frac{6 \tan^{2}{\left(\frac{x}{2} \right)}}{3 \tan^{3}{\left(\frac{x}{2} \right)} - 9 \tan^{2}{\left(\frac{x}{2} \right)} + 9 \tan{\left(\frac{x}{2} \right)} - 3} - \frac{2}{3 \tan^{3}{\left(\frac{x}{2} \right)} - 9 \tan^{2}{\left(\frac{x}{2} \right)} + 9 \tan{\left(\frac{x}{2} \right)} - 3}
Simplificar
Gráfica
0.000.050.100.150.200.250.300.350.400.450.500.550.600.650.700.75040
Trazar el gráfico f(x)
Respuesta [src] 
                                                                                         2                 
                                                                             /       ___\                  
  2                           2                                            6*\-1 + \/ 2 /                  
- - - ------------------------------------------------- - -------------------------------------------------
  3                       2                 3                                 2                 3          
              /       ___\      /       ___\        ___           /       ___\      /       ___\        ___
      -12 - 9*\-1 + \/ 2 /  + 3*\-1 + \/ 2 /  + 9*\/ 2    -12 - 9*\-1 + \/ 2 /  + 3*\-1 + \/ 2 /  + 9*\/ 2
−23−6(−1+2)2−12−9(−1+2)2+3(−1+2)3+92−2−12−9(−1+2)2+3(−1+2)3+92- \frac{2}{3} - \frac{6 \left(-1 + \sqrt{2}\right)^{2}}{-12 - 9 \left(-1 + \sqrt{2}\right)^{2} + 3 \left(-1 + \sqrt{2}\right)^{3} + 9 \sqrt{2}} - \frac{2}{-12 - 9 \left(-1 + \sqrt{2}\right)^{2} + 3 \left(-1 + \sqrt{2}\right)^{3} + 9 \sqrt{2}} 
=
= 
                                                                                         2                 
                                                                             /       ___\                  
  2                           2                                            6*\-1 + \/ 2 /                  
- - - ------------------------------------------------- - -------------------------------------------------
  3                       2                 3                                 2                 3          
              /       ___\      /       ___\        ___           /       ___\      /       ___\        ___
      -12 - 9*\-1 + \/ 2 /  + 3*\-1 + \/ 2 /  + 9*\/ 2    -12 - 9*\-1 + \/ 2 /  + 3*\-1 + \/ 2 /  + 9*\/ 2
−23−6(−1+2)2−12−9(−1+2)2+3(−1+2)3+92−2−12−9(−1+2)2+3(−1+2)3+92- \frac{2}{3} - \frac{6 \left(-1 + \sqrt{2}\right)^{2}}{-12 - 9 \left(-1 + \sqrt{2}\right)^{2} + 3 \left(-1 + \sqrt{2}\right)^{3} + 9 \sqrt{2}} - \frac{2}{-12 - 9 \left(-1 + \sqrt{2}\right)^{2} + 3 \left(-1 + \sqrt{2}\right)^{3} + 9 \sqrt{2}} 
-2/3 - 2/(-12 - 9*(-1 + sqrt(2))^2 + 3*(-1 + sqrt(2))^3 + 9*sqrt(2)) - 6*(-1 + sqrt(2))^2/(-12 - 9*(-1 + sqrt(2))^2 + 3*(-1 + sqrt(2))^3 + 9*sqrt(2))
Respuesta numérica [src] 
4.35702260395516
4.35702260395516

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

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