Sr Examen
Integral de 1/(2sinx+5cosx) dx
Solución
Ha introducido
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1 / | | 1 | ------------------- dx | 2*sin(x) + 5*cos(x) | / 0
$$\int\limits_{0}^{1} \frac{1}{2 \sin{\left(x \right)} + 5 \cos{\left(x \right)}}\, dx$$
Integral(1/(2*sin(x) + 5*cos(x)), (x, 0, 1))
Respuesta (Indefinida)
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/ ____ \ / ____ \ / ____ | 2 \/ 29 /x\| ____ | 2 \/ 29 /x\| | \/ 29 *log|- - - ------ + tan|-|| \/ 29 *log|- - + ------ + tan|-|| | 1 \ 5 5 \2// \ 5 5 \2// | ------------------- dx = C - --------------------------------- + --------------------------------- | 2*sin(x) + 5*cos(x) 29 29 | /
$$\int \frac{1}{2 \sin{\left(x \right)} + 5 \cos{\left(x \right)}}\, dx = C + \frac{\sqrt{29} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{2}{5} + \frac{\sqrt{29}}{5} \right)}}{29} - \frac{\sqrt{29} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{29}}{5} - \frac{2}{5} \right)}}{29}$$
Gráfica
Respuesta
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/ / ____\\ / ____\ / / ____\\ / ____ \
____ | |2 \/ 29 || ____ | 2 \/ 29 | ____ | |2 \/ 29 || ____ | 2 \/ 29 |
\/ 29 *|pi*I + log|- - tan(1/2) + ------|| \/ 29 *log|- - + ------| \/ 29 *|pi*I + log|- + ------|| \/ 29 *log|- - + ------ + tan(1/2)|
\ \5 5 // \ 5 5 / \ \5 5 // \ 5 5 /
- ------------------------------------------ - ------------------------ + ------------------------------- + -----------------------------------
29 29 29 29
$$\frac{\sqrt{29} \log{\left(- \frac{2}{5} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{29}}{5} \right)}}{29} - \frac{\sqrt{29} \log{\left(- \frac{2}{5} + \frac{\sqrt{29}}{5} \right)}}{29} - \frac{\sqrt{29} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{2}{5} + \frac{\sqrt{29}}{5} \right)} + i \pi\right)}{29} + \frac{\sqrt{29} \left(\log{\left(\frac{2}{5} + \frac{\sqrt{29}}{5} \right)} + i \pi\right)}{29}$$
=
=
/ / ____\\ / ____\ / / ____\\ / ____ \
____ | |2 \/ 29 || ____ | 2 \/ 29 | ____ | |2 \/ 29 || ____ | 2 \/ 29 |
\/ 29 *|pi*I + log|- - tan(1/2) + ------|| \/ 29 *log|- - + ------| \/ 29 *|pi*I + log|- + ------|| \/ 29 *log|- - + ------ + tan(1/2)|
\ \5 5 // \ 5 5 / \ \5 5 // \ 5 5 /
- ------------------------------------------ - ------------------------ + ------------------------------- + -----------------------------------
29 29 29 29
$$\frac{\sqrt{29} \log{\left(- \frac{2}{5} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{29}}{5} \right)}}{29} - \frac{\sqrt{29} \log{\left(- \frac{2}{5} + \frac{\sqrt{29}}{5} \right)}}{29} - \frac{\sqrt{29} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{2}{5} + \frac{\sqrt{29}}{5} \right)} + i \pi\right)}{29} + \frac{\sqrt{29} \left(\log{\left(\frac{2}{5} + \frac{\sqrt{29}}{5} \right)} + i \pi\right)}{29}$$
-sqrt(29)*(pi*i + log(2/5 - tan(1/2) + sqrt(29)/5))/29 - sqrt(29)*log(-2/5 + sqrt(29)/5)/29 + sqrt(29)*(pi*i + log(2/5 + sqrt(29)/5))/29 + sqrt(29)*log(-2/5 + sqrt(29)/5 + tan(1/2))/29
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.