Sr Examen
Integral de sin2x/(5-cos^2(2x)) dx
Solución
Ha introducido
[src]
1 / | | sin(2*x) | ------------- dx | 2 | 5 - cos (2*x) | / 0
$$\int\limits_{0}^{1} \frac{\sin{\left(2 x \right)}}{5 - \cos^{2}{\left(2 x \right)}}\, dx$$
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 / /
$$\int \frac{\sin{\left(2 x \right)}}{5 - \cos^{2}{\left(2 x \right)}}\, dx = C + \begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{2 \sqrt{5} \left(\cos^{2}{\left(x \right)} - \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(\cos^{2}{\left(x \right)} - \frac{1}{2}\right)^{2} > \frac{5}{4} \\- \frac{\sqrt{5} \operatorname{atanh}{\left(\frac{2 \sqrt{5} \left(\cos^{2}{\left(x \right)} - \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(\cos^{2}{\left(x \right)} - \frac{1}{2}\right)^{2} < \frac{5}{4} \end{cases}$$
Gráfica
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
$$- \frac{\sqrt{5} \log{\left(\cos{\left(2 \right)} + \sqrt{5} \right)}}{20} + \frac{\sqrt{5} \log{\left(1 + \sqrt{5} \right)}}{20} - \frac{\sqrt{5} \left(\log{\left(-1 + \sqrt{5} \right)} + i \pi\right)}{20} + \frac{\sqrt{5} \left(\log{\left(- \cos{\left(2 \right)} + \sqrt{5} \right)} + i \pi\right)}{20}$$
=
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___ / / ___\\ ___ / ___ \ ___ / / ___ \\ ___ / ___\
\/ 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
$$- \frac{\sqrt{5} \log{\left(\cos{\left(2 \right)} + \sqrt{5} \right)}}{20} + \frac{\sqrt{5} \log{\left(1 + \sqrt{5} \right)}}{20} - \frac{\sqrt{5} \left(\log{\left(-1 + \sqrt{5} \right)} + i \pi\right)}{20} + \frac{\sqrt{5} \left(\log{\left(- \cos{\left(2 \right)} + \sqrt{5} \right)} + i \pi\right)}{20}$$
-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
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.