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Resolución de integrales/ sin(x)/ cos(x)/ 1/(cos(x)+3*sin(x))

Integral de 1/(cos(x)+3*sin(x)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                     
  /                     
 |                      
 |          1           
 |  ----------------- dx
 |  cos(x) + 3*sin(x)   
 |                      
/                       
0
$$\int\limits_{0}^{1} \frac{1}{3 \sin{\left(x \right)} + \cos{\left(x \right)}}\, dx$$ 
Integral(1/(cos(x) + 3*sin(x)), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                             ____    /       ____      /x\\     ____    /       ____      /x\\
 |                            \/ 10 *log|-3 - \/ 10  + tan|-||   \/ 10 *log|-3 + \/ 10  + tan|-||
 |         1                            \                 \2//             \                 \2//
 | ----------------- dx = C - -------------------------------- + --------------------------------
 | cos(x) + 3*sin(x)                         10                                 10               
 |                                                                                               
/
$$\int \frac{1}{3 \sin{\left(x \right)} + \cos{\left(x \right)}}\, dx = C + \frac{\sqrt{10} \log{\left(\tan{\left(\frac{x}{2} \right)} - 3 + \sqrt{10} \right)}}{10} - \frac{\sqrt{10} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{10} - 3 \right)}}{10}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
    ____ /          /      ____           \\     ____    /       ____\     ____ /          /      ____\\     ____    /       ____           \
  \/ 10 *\pi*I + log\3 + \/ 10  - tan(1/2)//   \/ 10 *log\-3 + \/ 10 /   \/ 10 *\pi*I + log\3 + \/ 10 //   \/ 10 *log\-3 + \/ 10  + tan(1/2)/
- ------------------------------------------ - ----------------------- + ------------------------------- + ----------------------------------
                      10                                  10                            10                                 10
$$\frac{\sqrt{10} \log{\left(-3 + \tan{\left(\frac{1}{2} \right)} + \sqrt{10} \right)}}{10} - \frac{\sqrt{10} \log{\left(-3 + \sqrt{10} \right)}}{10} - \frac{\sqrt{10} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 3 + \sqrt{10} \right)} + i \pi\right)}{10} + \frac{\sqrt{10} \left(\log{\left(3 + \sqrt{10} \right)} + i \pi\right)}{10}$$ 
=
= 
    ____ /          /      ____           \\     ____    /       ____\     ____ /          /      ____\\     ____    /       ____           \
  \/ 10 *\pi*I + log\3 + \/ 10  - tan(1/2)//   \/ 10 *log\-3 + \/ 10 /   \/ 10 *\pi*I + log\3 + \/ 10 //   \/ 10 *log\-3 + \/ 10  + tan(1/2)/
- ------------------------------------------ - ----------------------- + ------------------------------- + ----------------------------------
                      10                                  10                            10                                 10
$$\frac{\sqrt{10} \log{\left(-3 + \tan{\left(\frac{1}{2} \right)} + \sqrt{10} \right)}}{10} - \frac{\sqrt{10} \log{\left(-3 + \sqrt{10} \right)}}{10} - \frac{\sqrt{10} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 3 + \sqrt{10} \right)} + i \pi\right)}{10} + \frac{\sqrt{10} \left(\log{\left(3 + \sqrt{10} \right)} + i \pi\right)}{10}$$ 
-sqrt(10)*(pi*i + log(3 + sqrt(10) - tan(1/2)))/10 - sqrt(10)*log(-3 + sqrt(10))/10 + sqrt(10)*(pi*i + log(3 + sqrt(10)))/10 + sqrt(10)*log(-3 + sqrt(10) + tan(1/2))/10
Respuesta numérica [src] 
0.495461101602369
0.495461101602369

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

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