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Resolución de integrales/ 3cosx/ cosx+1/ 4sinx/ 1/(4sinx+3cosx+1)

Integral de 1/(4sinx+3cosx+1) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                           
  /                           
 |                            
 |             1              
 |  ----------------------- dx
 |  4*sin(x) + 3*cos(x) + 1   
 |                            
/                             
0
$$\int\limits_{0}^{1} \frac{1}{\left(4 \sin{\left(x \right)} + 3 \cos{\left(x \right)}\right) + 1}\, dx$$ 
Integral(1/(4*sin(x) + 3*cos(x) + 1), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                   ___    /       ___      /x\\     ___    /       ___      /x\\
 |                                  \/ 6 *log|-2 - \/ 6  + tan|-||   \/ 6 *log|-2 + \/ 6  + tan|-||
 |            1                              \                \2//            \                \2//
 | ----------------------- dx = C - ------------------------------ + ------------------------------
 | 4*sin(x) + 3*cos(x) + 1                        12                               12              
 |                                                                                                 
/
$$\int \frac{1}{\left(4 \sin{\left(x \right)} + 3 \cos{\left(x \right)}\right) + 1}\, dx = C + \frac{\sqrt{6} \log{\left(\tan{\left(\frac{x}{2} \right)} - 2 + \sqrt{6} \right)}}{12} - \frac{\sqrt{6} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{6} - 2 \right)}}{12}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
    ___ /          /      ___           \\     ___    /       ___\     ___ /          /      ___\\     ___    /       ___           \
  \/ 6 *\pi*I + log\2 + \/ 6  - tan(1/2)//   \/ 6 *log\-2 + \/ 6 /   \/ 6 *\pi*I + log\2 + \/ 6 //   \/ 6 *log\-2 + \/ 6  + tan(1/2)/
- ---------------------------------------- - --------------------- + ----------------------------- + --------------------------------
                     12                                12                          12                               12
$$\frac{\sqrt{6} \log{\left(-2 + \tan{\left(\frac{1}{2} \right)} + \sqrt{6} \right)}}{12} - \frac{\sqrt{6} \log{\left(-2 + \sqrt{6} \right)}}{12} - \frac{\sqrt{6} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 2 + \sqrt{6} \right)} + i \pi\right)}{12} + \frac{\sqrt{6} \left(\log{\left(2 + \sqrt{6} \right)} + i \pi\right)}{12}$$ 
=
= 
    ___ /          /      ___           \\     ___    /       ___\     ___ /          /      ___\\     ___    /       ___           \
  \/ 6 *\pi*I + log\2 + \/ 6  - tan(1/2)//   \/ 6 *log\-2 + \/ 6 /   \/ 6 *\pi*I + log\2 + \/ 6 //   \/ 6 *log\-2 + \/ 6  + tan(1/2)/
- ---------------------------------------- - --------------------- + ----------------------------- + --------------------------------
                     12                                12                          12                               12
$$\frac{\sqrt{6} \log{\left(-2 + \tan{\left(\frac{1}{2} \right)} + \sqrt{6} \right)}}{12} - \frac{\sqrt{6} \log{\left(-2 + \sqrt{6} \right)}}{12} - \frac{\sqrt{6} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 2 + \sqrt{6} \right)} + i \pi\right)}{12} + \frac{\sqrt{6} \left(\log{\left(2 + \sqrt{6} \right)} + i \pi\right)}{12}$$ 
-sqrt(6)*(pi*i + log(2 + sqrt(6) - tan(1/2)))/12 - sqrt(6)*log(-2 + sqrt(6))/12 + sqrt(6)*(pi*i + log(2 + sqrt(6)))/12 + sqrt(6)*log(-2 + sqrt(6) + tan(1/2))/12
Respuesta numérica [src] 
0.189105009244451
0.189105009244451

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

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