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Resolución de integrales/ cos(x)/ sin(x)/ cos(2*x)/ cos(2*x)/(cos(x)+sin(x))

Integral de cos(2*x)/(cos(x)+sin(x)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                   
  /                   
 |                    
 |      cos(2*x)      
 |  --------------- dx
 |  cos(x) + sin(x)   
 |                    
/                     
0
$$\int\limits_{0}^{1} \frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}\, dx$$ 
Integral(cos(2*x)/(cos(x) + sin(x)), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                           ___    /       ___      /x\\           /x\       ___    /       ___      /x\\       ___    /       ___      /x\\       ___    /       ___      /x\\       ___    2/x\    /       ___      /x\\       ___    2/x\    /       ___      /x\\
 |                                          \/ 2 *log|-1 - \/ 2  + tan|-||      8*tan|-|     \/ 2 *log|-1 + \/ 2  + tan|-||   2*\/ 2 *log|-1 - \/ 2  + tan|-||   2*\/ 2 *log|-1 + \/ 2  + tan|-||   2*\/ 2 *tan |-|*log|-1 - \/ 2  + tan|-||   2*\/ 2 *tan |-|*log|-1 + \/ 2  + tan|-||
 |     cos(2*x)                   8                  \                \2//           \2/              \                \2//              \                \2//              \                \2//               \2/    \                \2//               \2/    \                \2//
 | --------------- dx = C + ------------- + ------------------------------ + ------------- - ------------------------------ - -------------------------------- + -------------------------------- - ---------------------------------------- + ----------------------------------------
 | cos(x) + sin(x)                   2/x\                 2                           2/x\                 2                                    2/x\                               2/x\                                   2/x\                                       2/x\              
 |                          4 + 4*tan |-|                                    4 + 4*tan |-|                                             4 + 4*tan |-|                      4 + 4*tan |-|                          4 + 4*tan |-|                              4 + 4*tan |-|              
/                                     \2/                                              \2/                                                       \2/                                \2/                                    \2/                                        \2/
$$\int \frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}\, dx = C - \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 + \sqrt{2} \right)}}{2} + \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{2} - 1 \right)}}{2} + \frac{2 \sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 + \sqrt{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{2 \sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 + \sqrt{2} \right)}}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{2 \sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{2} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{2 \sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{2} - 1 \right)}}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{8 \tan{\left(\frac{x}{2} \right)}}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{8}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                         ___ /          /      ___           \\                       ___    /       ___           \       ___ /          /      ___           \\       ___    /       ___           \       ___    2      /          /      ___           \\       ___    2         /       ___           \
            8          \/ 2 *\pi*I + log\1 + \/ 2  - tan(1/2)//      8*tan(1/2)     \/ 2 *log\-1 + \/ 2  + tan(1/2)/   2*\/ 2 *\pi*I + log\1 + \/ 2  - tan(1/2)//   2*\/ 2 *log\-1 + \/ 2  + tan(1/2)/   2*\/ 2 *tan (1/2)*\pi*I + log\1 + \/ 2  - tan(1/2)//   2*\/ 2 *tan (1/2)*log\-1 + \/ 2  + tan(1/2)/
-2 + --------------- + ---------------------------------------- + --------------- - -------------------------------- - ------------------------------------------ + ---------------------------------- - ---------------------------------------------------- + --------------------------------------------
              2                           2                                2                       2                                         2                                        2                                             2                                                  2                    
     4 + 4*tan (1/2)                                              4 + 4*tan (1/2)                                                   4 + 4*tan (1/2)                          4 + 4*tan (1/2)                               4 + 4*tan (1/2)                                    4 + 4*tan (1/2)
$$-2 + \frac{2 \sqrt{2} \log{\left(-1 + \tan{\left(\frac{1}{2} \right)} + \sqrt{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{2 \sqrt{2} \log{\left(-1 + \tan{\left(\frac{1}{2} \right)} + \sqrt{2} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{\sqrt{2} \log{\left(-1 + \tan{\left(\frac{1}{2} \right)} + \sqrt{2} \right)}}{2} + \frac{8 \tan{\left(\frac{1}{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{8}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{2 \sqrt{2} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 1 + \sqrt{2} \right)} + i \pi\right)}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{2 \sqrt{2} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 1 + \sqrt{2} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{\sqrt{2} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 1 + \sqrt{2} \right)} + i \pi\right)}{2}$$ 
=
= 
                         ___ /          /      ___           \\                       ___    /       ___           \       ___ /          /      ___           \\       ___    /       ___           \       ___    2      /          /      ___           \\       ___    2         /       ___           \
            8          \/ 2 *\pi*I + log\1 + \/ 2  - tan(1/2)//      8*tan(1/2)     \/ 2 *log\-1 + \/ 2  + tan(1/2)/   2*\/ 2 *\pi*I + log\1 + \/ 2  - tan(1/2)//   2*\/ 2 *log\-1 + \/ 2  + tan(1/2)/   2*\/ 2 *tan (1/2)*\pi*I + log\1 + \/ 2  - tan(1/2)//   2*\/ 2 *tan (1/2)*log\-1 + \/ 2  + tan(1/2)/
-2 + --------------- + ---------------------------------------- + --------------- - -------------------------------- - ------------------------------------------ + ---------------------------------- - ---------------------------------------------------- + --------------------------------------------
              2                           2                                2                       2                                         2                                        2                                             2                                                  2                    
     4 + 4*tan (1/2)                                              4 + 4*tan (1/2)                                                   4 + 4*tan (1/2)                          4 + 4*tan (1/2)                               4 + 4*tan (1/2)                                    4 + 4*tan (1/2)
$$-2 + \frac{2 \sqrt{2} \log{\left(-1 + \tan{\left(\frac{1}{2} \right)} + \sqrt{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{2 \sqrt{2} \log{\left(-1 + \tan{\left(\frac{1}{2} \right)} + \sqrt{2} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{\sqrt{2} \log{\left(-1 + \tan{\left(\frac{1}{2} \right)} + \sqrt{2} \right)}}{2} + \frac{8 \tan{\left(\frac{1}{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{8}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{2 \sqrt{2} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 1 + \sqrt{2} \right)} + i \pi\right)}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{2 \sqrt{2} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 1 + \sqrt{2} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{\sqrt{2} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 1 + \sqrt{2} \right)} + i \pi\right)}{2}$$ 
-2 + 8/(4 + 4*tan(1/2)^2) + sqrt(2)*(pi*i + log(1 + sqrt(2) - tan(1/2)))/2 + 8*tan(1/2)/(4 + 4*tan(1/2)^2) - sqrt(2)*log(-1 + sqrt(2) + tan(1/2))/2 - 2*sqrt(2)*(pi*i + log(1 + sqrt(2) - tan(1/2)))/(4 + 4*tan(1/2)^2) + 2*sqrt(2)*log(-1 + sqrt(2) + tan(1/2))/(4 + 4*tan(1/2)^2) - 2*sqrt(2)*tan(1/2)^2*(pi*i + log(1 + sqrt(2) - tan(1/2)))/(4 + 4*tan(1/2)^2) + 2*sqrt(2)*tan(1/2)^2*log(-1 + sqrt(2) + tan(1/2))/(4 + 4*tan(1/2)^2)
Respuesta numérica [src] 
0.381773290676036
0.381773290676036

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

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