Sr Examen
Integral de 1/(2sinx+3cosx) dx
Solución
Ha introducido
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1 / | | 1 | ------------------- dx | 2*sin(x) + 3*cos(x) | / 0
$$\int\limits_{0}^{1} \frac{1}{2 \sin{\left(x \right)} + 3 \cos{\left(x \right)}}\, dx$$
Integral(1/(2*sin(x) + 3*cos(x)), (x, 0, 1))
Respuesta (Indefinida)
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/ ____ \ / ____ \ / ____ | 2 \/ 13 /x\| ____ | 2 \/ 13 /x\| | \/ 13 *log|- - - ------ + tan|-|| \/ 13 *log|- - + ------ + tan|-|| | 1 \ 3 3 \2// \ 3 3 \2// | ------------------- dx = C - --------------------------------- + --------------------------------- | 2*sin(x) + 3*cos(x) 13 13 | /
$$\int \frac{1}{2 \sin{\left(x \right)} + 3 \cos{\left(x \right)}}\, dx = C + \frac{\sqrt{13} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{2}{3} + \frac{\sqrt{13}}{3} \right)}}{13} - \frac{\sqrt{13} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{13}}{3} - \frac{2}{3} \right)}}{13}$$
Gráfica
Respuesta
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/ / ____\\ / ____\ / / ____\\ / ____ \
____ | |2 \/ 13 || ____ | 2 \/ 13 | ____ | |2 \/ 13 || ____ | 2 \/ 13 |
\/ 13 *|pi*I + log|- - tan(1/2) + ------|| \/ 13 *log|- - + ------| \/ 13 *|pi*I + log|- + ------|| \/ 13 *log|- - + ------ + tan(1/2)|
\ \3 3 // \ 3 3 / \ \3 3 // \ 3 3 /
- ------------------------------------------ - ------------------------ + ------------------------------- + -----------------------------------
13 13 13 13
$$\frac{\sqrt{13} \log{\left(- \frac{2}{3} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{13}}{3} \right)}}{13} - \frac{\sqrt{13} \log{\left(- \frac{2}{3} + \frac{\sqrt{13}}{3} \right)}}{13} - \frac{\sqrt{13} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{2}{3} + \frac{\sqrt{13}}{3} \right)} + i \pi\right)}{13} + \frac{\sqrt{13} \left(\log{\left(\frac{2}{3} + \frac{\sqrt{13}}{3} \right)} + i \pi\right)}{13}$$
=
=
/ / ____\\ / ____\ / / ____\\ / ____ \
____ | |2 \/ 13 || ____ | 2 \/ 13 | ____ | |2 \/ 13 || ____ | 2 \/ 13 |
\/ 13 *|pi*I + log|- - tan(1/2) + ------|| \/ 13 *log|- - + ------| \/ 13 *|pi*I + log|- + ------|| \/ 13 *log|- - + ------ + tan(1/2)|
\ \3 3 // \ 3 3 / \ \3 3 // \ 3 3 /
- ------------------------------------------ - ------------------------ + ------------------------------- + -----------------------------------
13 13 13 13
$$\frac{\sqrt{13} \log{\left(- \frac{2}{3} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{13}}{3} \right)}}{13} - \frac{\sqrt{13} \log{\left(- \frac{2}{3} + \frac{\sqrt{13}}{3} \right)}}{13} - \frac{\sqrt{13} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{2}{3} + \frac{\sqrt{13}}{3} \right)} + i \pi\right)}{13} + \frac{\sqrt{13} \left(\log{\left(\frac{2}{3} + \frac{\sqrt{13}}{3} \right)} + i \pi\right)}{13}$$
-sqrt(13)*(pi*i + log(2/3 - tan(1/2) + sqrt(13)/3))/13 - sqrt(13)*log(-2/3 + sqrt(13)/3)/13 + sqrt(13)*(pi*i + log(2/3 + sqrt(13)/3))/13 + sqrt(13)*log(-2/3 + sqrt(13)/3 + tan(1/2))/13
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.