Sr Examen
Integral de 1/(2-4sinx+7cosx) dx
Solución
Ha introducido
[src]
1 / | | 1 | ----------------------- dx | 2 - 4*sin(x) + 7*cos(x) | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(2 - 4 \sin{\left(x \right)}\right) + 7 \cos{\left(x \right)}}\, dx$$
Integral(1/(2 - 4*sin(x) + 7*cos(x)), (x, 0, 1))
Respuesta (Indefinida)
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/ ____ \ / ____ \ / ____ |4 \/ 61 /x\| ____ |4 \/ 61 /x\| | \/ 61 *log|- - ------ + tan|-|| \/ 61 *log|- + ------ + tan|-|| | 1 \5 5 \2// \5 5 \2// | ----------------------- dx = C - ------------------------------- + ------------------------------- | 2 - 4*sin(x) + 7*cos(x) 61 61 | /
$$\int \frac{1}{\left(2 - 4 \sin{\left(x \right)}\right) + 7 \cos{\left(x \right)}}\, dx = C + \frac{\sqrt{61} \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{4}{5} + \frac{\sqrt{61}}{5} \right)}}{61} - \frac{\sqrt{61} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{61}}{5} + \frac{4}{5} \right)}}{61}$$
Gráfica
Respuesta
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/ / ____\\ / ____\ / / ____\\ / ____ \
____ | | 4 \/ 61 || ____ |4 \/ 61 | ____ | | 4 \/ 61 || ____ |4 \/ 61 |
\/ 61 *|pi*I + log|- - - tan(1/2) + ------|| \/ 61 *log|- + ------| \/ 61 *|pi*I + log|- - + ------|| \/ 61 *log|- + ------ + tan(1/2)|
\ \ 5 5 // \5 5 / \ \ 5 5 // \5 5 /
- -------------------------------------------- - ---------------------- + --------------------------------- + ---------------------------------
61 61 61 61
$$- \frac{\sqrt{61} \log{\left(\frac{4}{5} + \frac{\sqrt{61}}{5} \right)}}{61} + \frac{\sqrt{61} \log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{4}{5} + \frac{\sqrt{61}}{5} \right)}}{61} - \frac{\sqrt{61} \left(\log{\left(- \frac{4}{5} - \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{61}}{5} \right)} + i \pi\right)}{61} + \frac{\sqrt{61} \left(\log{\left(- \frac{4}{5} + \frac{\sqrt{61}}{5} \right)} + i \pi\right)}{61}$$
=
=
/ / ____\\ / ____\ / / ____\\ / ____ \
____ | | 4 \/ 61 || ____ |4 \/ 61 | ____ | | 4 \/ 61 || ____ |4 \/ 61 |
\/ 61 *|pi*I + log|- - - tan(1/2) + ------|| \/ 61 *log|- + ------| \/ 61 *|pi*I + log|- - + ------|| \/ 61 *log|- + ------ + tan(1/2)|
\ \ 5 5 // \5 5 / \ \ 5 5 // \5 5 /
- -------------------------------------------- - ---------------------- + --------------------------------- + ---------------------------------
61 61 61 61
$$- \frac{\sqrt{61} \log{\left(\frac{4}{5} + \frac{\sqrt{61}}{5} \right)}}{61} + \frac{\sqrt{61} \log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{4}{5} + \frac{\sqrt{61}}{5} \right)}}{61} - \frac{\sqrt{61} \left(\log{\left(- \frac{4}{5} - \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{61}}{5} \right)} + i \pi\right)}{61} + \frac{\sqrt{61} \left(\log{\left(- \frac{4}{5} + \frac{\sqrt{61}}{5} \right)} + i \pi\right)}{61}$$
-sqrt(61)*(pi*i + log(-4/5 - tan(1/2) + sqrt(61)/5))/61 - sqrt(61)*log(4/5 + sqrt(61)/5)/61 + sqrt(61)*(pi*i + log(-4/5 + sqrt(61)/5))/61 + sqrt(61)*log(4/5 + sqrt(61)/5 + tan(1/2))/61
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.