Sr Examen
Integral de 1/(6cosx+2sinx-3) dx
Solución
Ha introducido
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1 / | | 1 | ----------------------- dx | 6*cos(x) + 2*sin(x) - 3 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(2 \sin{\left(x \right)} + 6 \cos{\left(x \right)}\right) - 3}\, dx$$
Integral(1/(6*cos(x) + 2*sin(x) - 3), (x, 0, 1))
Respuesta (Indefinida)
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/ ____ \ / ____ \ / ____ | 2 \/ 31 /x\| ____ | 2 \/ 31 /x\| | \/ 31 *log|- - - ------ + tan|-|| \/ 31 *log|- - + ------ + tan|-|| | 1 \ 9 9 \2// \ 9 9 \2// | ----------------------- dx = C - --------------------------------- + --------------------------------- | 6*cos(x) + 2*sin(x) - 3 31 31 | /
$$\int \frac{1}{\left(2 \sin{\left(x \right)} + 6 \cos{\left(x \right)}\right) - 3}\, dx = C + \frac{\sqrt{31} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{2}{9} + \frac{\sqrt{31}}{9} \right)}}{31} - \frac{\sqrt{31} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{31}}{9} - \frac{2}{9} \right)}}{31}$$
Gráfica
Respuesta
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/ / ____\\ / ____\ / / ____\\ / ____ \
____ | |2 \/ 31 || ____ | 2 \/ 31 | ____ | |2 \/ 31 || ____ | 2 \/ 31 |
\/ 31 *|pi*I + log|- - tan(1/2) + ------|| \/ 31 *log|- - + ------| \/ 31 *|pi*I + log|- + ------|| \/ 31 *log|- - + ------ + tan(1/2)|
\ \9 9 // \ 9 9 / \ \9 9 // \ 9 9 /
- ------------------------------------------ - ------------------------ + ------------------------------- + -----------------------------------
31 31 31 31
$$\frac{\sqrt{31} \log{\left(- \frac{2}{9} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{31}}{9} \right)}}{31} - \frac{\sqrt{31} \log{\left(- \frac{2}{9} + \frac{\sqrt{31}}{9} \right)}}{31} - \frac{\sqrt{31} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{2}{9} + \frac{\sqrt{31}}{9} \right)} + i \pi\right)}{31} + \frac{\sqrt{31} \left(\log{\left(\frac{2}{9} + \frac{\sqrt{31}}{9} \right)} + i \pi\right)}{31}$$
=
=
/ / ____\\ / ____\ / / ____\\ / ____ \
____ | |2 \/ 31 || ____ | 2 \/ 31 | ____ | |2 \/ 31 || ____ | 2 \/ 31 |
\/ 31 *|pi*I + log|- - tan(1/2) + ------|| \/ 31 *log|- - + ------| \/ 31 *|pi*I + log|- + ------|| \/ 31 *log|- - + ------ + tan(1/2)|
\ \9 9 // \ 9 9 / \ \9 9 // \ 9 9 /
- ------------------------------------------ - ------------------------ + ------------------------------- + -----------------------------------
31 31 31 31
$$\frac{\sqrt{31} \log{\left(- \frac{2}{9} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{31}}{9} \right)}}{31} - \frac{\sqrt{31} \log{\left(- \frac{2}{9} + \frac{\sqrt{31}}{9} \right)}}{31} - \frac{\sqrt{31} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{2}{9} + \frac{\sqrt{31}}{9} \right)} + i \pi\right)}{31} + \frac{\sqrt{31} \left(\log{\left(\frac{2}{9} + \frac{\sqrt{31}}{9} \right)} + i \pi\right)}{31}$$
-sqrt(31)*(pi*i + log(2/9 - tan(1/2) + sqrt(31)/9))/31 - sqrt(31)*log(-2/9 + sqrt(31)/9)/31 + sqrt(31)*(pi*i + log(2/9 + sqrt(31)/9))/31 + sqrt(31)*log(-2/9 + sqrt(31)/9 + tan(1/2))/31
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.