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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                   
  /                   
 |                    
 |   sin(x)*cos(x)    
 |  --------------- dx
 |  sin(x) + cos(x)   
 |                    
/                     
0
$$\int\limits_{0}^{1} \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}\, dx$$ 
Integral((sin(x)*cos(x))/(sin(x) + cos(x)), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                 /x\       ___    /       ___      /x\\     ___    /       ___      /x\\     ___    2/x\    /       ___      /x\\     ___    2/x\    /       ___      /x\\
 |                                             4*tan|-|     \/ 2 *log|-1 - \/ 2  + tan|-||   \/ 2 *log|-1 + \/ 2  + tan|-||   \/ 2 *tan |-|*log|-1 - \/ 2  + tan|-||   \/ 2 *tan |-|*log|-1 + \/ 2  + tan|-||
 |  sin(x)*cos(x)                 4                 \2/              \                \2//            \                \2//             \2/    \                \2//             \2/    \                \2//
 | --------------- dx = C - ------------- + ------------- + ------------------------------ - ------------------------------ + -------------------------------------- - --------------------------------------
 | sin(x) + cos(x)                   2/x\            2/x\                    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/
$$\int \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}\, dx = C - \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 + \sqrt{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 + \sqrt{2} \right)}}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{2} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{2} - 1 \right)}}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{4 \tan{\left(\frac{x}{2} \right)}}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{4}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                                          ___ /          /      ___\\     ___    /       ___\     ___ /          /      ___           \\     ___    /       ___           \     ___    2      /          /      ___           \\     ___    2         /       ___           \
           4             4*tan(1/2)     \/ 2 *\pi*I + log\1 + \/ 2 //   \/ 2 *log\-1 + \/ 2 /   \/ 2 *\pi*I + log\1 + \/ 2  - tan(1/2)//   \/ 2 *log\-1 + \/ 2  + tan(1/2)/   \/ 2 *tan (1/2)*\pi*I + log\1 + \/ 2  - tan(1/2)//   \/ 2 *tan (1/2)*log\-1 + \/ 2  + tan(1/2)/
1 - --------------- + --------------- - ----------------------------- + --------------------- + ---------------------------------------- - -------------------------------- + -------------------------------------------------- - ------------------------------------------
             2                 2                      4                           4                                  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)
$$- \frac{4}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{\sqrt{2} \log{\left(-1 + \sqrt{2} \right)}}{4} - \frac{\sqrt{2} \log{\left(-1 + \tan{\left(\frac{1}{2} \right)} + \sqrt{2} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{\sqrt{2} \log{\left(-1 + \tan{\left(\frac{1}{2} \right)} + \sqrt{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{4 \tan{\left(\frac{1}{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + 1 - \frac{\sqrt{2} \left(\log{\left(1 + \sqrt{2} \right)} + i \pi\right)}{4} + \frac{\sqrt{2} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 1 + \sqrt{2} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{\sqrt{2} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 1 + \sqrt{2} \right)} + i \pi\right)}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4}$$ 
=
= 
                                          ___ /          /      ___\\     ___    /       ___\     ___ /          /      ___           \\     ___    /       ___           \     ___    2      /          /      ___           \\     ___    2         /       ___           \
           4             4*tan(1/2)     \/ 2 *\pi*I + log\1 + \/ 2 //   \/ 2 *log\-1 + \/ 2 /   \/ 2 *\pi*I + log\1 + \/ 2  - tan(1/2)//   \/ 2 *log\-1 + \/ 2  + tan(1/2)/   \/ 2 *tan (1/2)*\pi*I + log\1 + \/ 2  - tan(1/2)//   \/ 2 *tan (1/2)*log\-1 + \/ 2  + tan(1/2)/
1 - --------------- + --------------- - ----------------------------- + --------------------- + ---------------------------------------- - -------------------------------- + -------------------------------------------------- - ------------------------------------------
             2                 2                      4                           4                                  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)
$$- \frac{4}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{\sqrt{2} \log{\left(-1 + \sqrt{2} \right)}}{4} - \frac{\sqrt{2} \log{\left(-1 + \tan{\left(\frac{1}{2} \right)} + \sqrt{2} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{\sqrt{2} \log{\left(-1 + \tan{\left(\frac{1}{2} \right)} + \sqrt{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{4 \tan{\left(\frac{1}{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + 1 - \frac{\sqrt{2} \left(\log{\left(1 + \sqrt{2} \right)} + i \pi\right)}{4} + \frac{\sqrt{2} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 1 + \sqrt{2} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{\sqrt{2} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 1 + \sqrt{2} \right)} + i \pi\right)}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4}$$ 
1 - 4/(4 + 4*tan(1/2)^2) + 4*tan(1/2)/(4 + 4*tan(1/2)^2) - sqrt(2)*(pi*i + log(1 + sqrt(2)))/4 + sqrt(2)*log(-1 + sqrt(2))/4 + sqrt(2)*(pi*i + log(1 + sqrt(2) - tan(1/2)))/(4 + 4*tan(1/2)^2) - sqrt(2)*log(-1 + sqrt(2) + tan(1/2))/(4 + 4*tan(1/2)^2) + sqrt(2)*tan(1/2)^2*(pi*i + log(1 + sqrt(2) - tan(1/2)))/(4 + 4*tan(1/2)^2) - sqrt(2)*tan(1/2)^2*log(-1 + sqrt(2) + tan(1/2))/(4 + 4*tan(1/2)^2)
Respuesta numérica [src] 
0.262509339440237
0.262509339440237

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

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