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Resolución de integrales/ cos^2/ sin2x/ sin2x/(5-cos^2(2x))

Integral de sin2x/(5-cos^2(2x)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                 
  /                 
 |                  
 |     sin(2*x)     
 |  ------------- dx
 |         2        
 |  5 - cos (2*x)   
 |                  
/                   
0
∫01sin⁡(2x)5−cos⁡2(2x) dx\int\limits_{0}^{1} \frac{\sin{\left(2 x \right)}}{5 - \cos^{2}{\left(2 x \right)}}\, dx 
Integral(sin(2*x)/(5 - cos(2*x)^2), (x, 0, 1))
Respuesta (Indefinida) [src] 
                          //            /    ___ /  1      2   \\                             \
                          ||            |2*\/ 5 *|- - + cos (x)||                             |
                          ||   ___      |        \  2          /|                             |
                          ||-\/ 5 *acoth|-----------------------|                      2      |
  /                       ||            \           5           /       /  1      2   \       |
 |                        ||--------------------------------------  for |- - + cos (x)|  > 5/4|
 |    sin(2*x)            ||                  10                        \  2          /       |
 | ------------- dx = C + |<                                                                  |
 |        2               ||            /    ___ /  1      2   \\                             |
 | 5 - cos (2*x)          ||            |2*\/ 5 *|- - + cos (x)||                             |
 |                        ||   ___      |        \  2          /|                             |
/                         ||-\/ 5 *atanh|-----------------------|                      2      |
                          ||            \           5           /       /  1      2   \       |
                          ||--------------------------------------  for |- - + cos (x)|  < 5/4|
                          \\                  10                        \  2          /       /
∫sin⁡(2x)5−cos⁡2(2x) dx=C+{−5acoth⁡(25(cos⁡2(x)−12)5)10for (cos⁡2(x)−12)2>54−5atanh⁡(25(cos⁡2(x)−12)5)10for (cos⁡2(x)−12)2<54\int \frac{\sin{\left(2 x \right)}}{5 - \cos^{2}{\left(2 x \right)}}\, dx = C + \begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{2 \sqrt{5} \left(\cos^{2}{\left(x \right)} - \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(\cos^{2}{\left(x \right)} - \frac{1}{2}\right)^{2} > \frac{5}{4} \\- \frac{\sqrt{5} \operatorname{atanh}{\left(\frac{2 \sqrt{5} \left(\cos^{2}{\left(x \right)} - \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(\cos^{2}{\left(x \right)} - \frac{1}{2}\right)^{2} < \frac{5}{4} \end{cases}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.900.00.4
Trazar el gráfico f(x)
Respuesta [src] 
    ___ /          /       ___\\     ___    /  ___         \     ___ /          /  ___         \\     ___    /      ___\
  \/ 5 *\pi*I + log\-1 + \/ 5 //   \/ 5 *log\\/ 5  + cos(2)/   \/ 5 *\pi*I + log\\/ 5  - cos(2)//   \/ 5 *log\1 + \/ 5 /
- ------------------------------ - ------------------------- + ---------------------------------- + --------------------
                20                             20                              20                            20
−5log⁡(cos⁡(2)+5)20+5log⁡(1+5)20−5(log⁡(−1+5)+iπ)20+5(log⁡(−cos⁡(2)+5)+iπ)20- \frac{\sqrt{5} \log{\left(\cos{\left(2 \right)} + \sqrt{5} \right)}}{20} + \frac{\sqrt{5} \log{\left(1 + \sqrt{5} \right)}}{20} - \frac{\sqrt{5} \left(\log{\left(-1 + \sqrt{5} \right)} + i \pi\right)}{20} + \frac{\sqrt{5} \left(\log{\left(- \cos{\left(2 \right)} + \sqrt{5} \right)} + i \pi\right)}{20} 
=
= 
    ___ /          /       ___\\     ___    /  ___         \     ___ /          /  ___         \\     ___    /      ___\
  \/ 5 *\pi*I + log\-1 + \/ 5 //   \/ 5 *log\\/ 5  + cos(2)/   \/ 5 *\pi*I + log\\/ 5  - cos(2)//   \/ 5 *log\1 + \/ 5 /
- ------------------------------ - ------------------------- + ---------------------------------- + --------------------
                20                             20                              20                            20
−5log⁡(cos⁡(2)+5)20+5log⁡(1+5)20−5(log⁡(−1+5)+iπ)20+5(log⁡(−cos⁡(2)+5)+iπ)20- \frac{\sqrt{5} \log{\left(\cos{\left(2 \right)} + \sqrt{5} \right)}}{20} + \frac{\sqrt{5} \log{\left(1 + \sqrt{5} \right)}}{20} - \frac{\sqrt{5} \left(\log{\left(-1 + \sqrt{5} \right)} + i \pi\right)}{20} + \frac{\sqrt{5} \left(\log{\left(- \cos{\left(2 \right)} + \sqrt{5} \right)} + i \pi\right)}{20} 
-sqrt(5)*(pi*i + log(-1 + sqrt(5)))/20 - sqrt(5)*log(sqrt(5) + cos(2))/20 + sqrt(5)*(pi*i + log(sqrt(5) - cos(2)))/20 + sqrt(5)*log(1 + sqrt(5))/20
Respuesta numérica [src] 
0.149707607550958
0.149707607550958

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