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Resolución de integrales/ x+cosx/ 1÷(sinx+cosx)

Integral de 1÷(sinx+cosx) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                   
  /                   
 |                    
 |         1          
 |  --------------- dx
 |  sin(x) + cos(x)   
 |                    
/                     
0
$$\int\limits_{0}^{1} \frac{1}{\sin{\left(x \right)} + \cos{\left(x \right)}}\, dx$$ 
Integral(1/(sin(x) + cos(x)), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                           ___    /       ___      /x\\     ___    /       ___      /x\\
 |                          \/ 2 *log|-1 + \/ 2  + tan|-||   \/ 2 *log|-1 - \/ 2  + tan|-||
 |        1                          \                \2//            \                \2//
 | --------------- dx = C + ------------------------------ - ------------------------------
 | sin(x) + cos(x)                        2                                2               
 |                                                                                         
/
$$\int \frac{1}{\sin{\left(x \right)} + \cos{\left(x \right)}}\, dx = C + \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 + \sqrt{2} \right)}}{2} - \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{2} - 1 \right)}}{2}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
  ___ /          /      ___\\     ___    /       ___           \     ___ /          /      ___           \\     ___    /       ___\
\/ 2 *\pi*I + log\1 + \/ 2 //   \/ 2 *log\-1 + \/ 2  + tan(1/2)/   \/ 2 *\pi*I + log\1 + \/ 2  - tan(1/2)//   \/ 2 *log\-1 + \/ 2 /
----------------------------- + -------------------------------- - ---------------------------------------- - ---------------------
              2                                2                                      2                                 2
$$\frac{\sqrt{2} \log{\left(-1 + \tan{\left(\frac{1}{2} \right)} + \sqrt{2} \right)}}{2} - \frac{\sqrt{2} \log{\left(-1 + \sqrt{2} \right)}}{2} - \frac{\sqrt{2} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 1 + \sqrt{2} \right)} + i \pi\right)}{2} + \frac{\sqrt{2} \left(\log{\left(1 + \sqrt{2} \right)} + i \pi\right)}{2}$$ 
=
= 
  ___ /          /      ___\\     ___    /       ___           \     ___ /          /      ___           \\     ___    /       ___\
\/ 2 *\pi*I + log\1 + \/ 2 //   \/ 2 *log\-1 + \/ 2  + tan(1/2)/   \/ 2 *\pi*I + log\1 + \/ 2  - tan(1/2)//   \/ 2 *log\-1 + \/ 2 /
----------------------------- + -------------------------------- - ---------------------------------------- - ---------------------
              2                                2                                      2                                 2
$$\frac{\sqrt{2} \log{\left(-1 + \tan{\left(\frac{1}{2} \right)} + \sqrt{2} \right)}}{2} - \frac{\sqrt{2} \log{\left(-1 + \sqrt{2} \right)}}{2} - \frac{\sqrt{2} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 1 + \sqrt{2} \right)} + i \pi\right)}{2} + \frac{\sqrt{2} \left(\log{\left(1 + \sqrt{2} \right)} + i \pi\right)}{2}$$ 
sqrt(2)*(pi*i + log(1 + sqrt(2)))/2 + sqrt(2)*log(-1 + sqrt(2) + tan(1/2))/2 - sqrt(2)*(pi*i + log(1 + sqrt(2) - tan(1/2)))/2 - sqrt(2)*log(-1 + sqrt(2))/2
Respuesta numérica [src] 
0.776150000059282
0.776150000059282

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

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