Sr Examen
Integral de (Sinx+6cosx)^-1 dx
Solución
Ha introducido
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1 / | | 1 | ----------------- dx | sin(x) + 6*cos(x) | / 0
$$\int\limits_{0}^{1} \frac{1}{\sin{\left(x \right)} + 6 \cos{\left(x \right)}}\, dx$$
Integral(1/(sin(x) + 6*cos(x)), (x, 0, 1))
Respuesta (Indefinida)
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/ ____ \ / ____ \ / ____ | 1 \/ 37 /x\| ____ | 1 \/ 37 /x\| | \/ 37 *log|- - - ------ + tan|-|| \/ 37 *log|- - + ------ + tan|-|| | 1 \ 6 6 \2// \ 6 6 \2// | ----------------- dx = C - --------------------------------- + --------------------------------- | sin(x) + 6*cos(x) 37 37 | /
$$\int \frac{1}{\sin{\left(x \right)} + 6 \cos{\left(x \right)}}\, dx = C + \frac{\sqrt{37} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{1}{6} + \frac{\sqrt{37}}{6} \right)}}{37} - \frac{\sqrt{37} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{37}}{6} - \frac{1}{6} \right)}}{37}$$
Gráfica
Respuesta
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/ / ____\\ / ____\ / / ____\\ / ____ \
____ | |1 \/ 37 || ____ | 1 \/ 37 | ____ | |1 \/ 37 || ____ | 1 \/ 37 |
\/ 37 *|pi*I + log|- - tan(1/2) + ------|| \/ 37 *log|- - + ------| \/ 37 *|pi*I + log|- + ------|| \/ 37 *log|- - + ------ + tan(1/2)|
\ \6 6 // \ 6 6 / \ \6 6 // \ 6 6 /
- ------------------------------------------ - ------------------------ + ------------------------------- + -----------------------------------
37 37 37 37
$$- \frac{\sqrt{37} \log{\left(- \frac{1}{6} + \frac{\sqrt{37}}{6} \right)}}{37} + \frac{\sqrt{37} \log{\left(- \frac{1}{6} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{37}}{6} \right)}}{37} - \frac{\sqrt{37} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{1}{6} + \frac{\sqrt{37}}{6} \right)} + i \pi\right)}{37} + \frac{\sqrt{37} \left(\log{\left(\frac{1}{6} + \frac{\sqrt{37}}{6} \right)} + i \pi\right)}{37}$$
=
=
/ / ____\\ / ____\ / / ____\\ / ____ \
____ | |1 \/ 37 || ____ | 1 \/ 37 | ____ | |1 \/ 37 || ____ | 1 \/ 37 |
\/ 37 *|pi*I + log|- - tan(1/2) + ------|| \/ 37 *log|- - + ------| \/ 37 *|pi*I + log|- + ------|| \/ 37 *log|- - + ------ + tan(1/2)|
\ \6 6 // \ 6 6 / \ \6 6 // \ 6 6 /
- ------------------------------------------ - ------------------------ + ------------------------------- + -----------------------------------
37 37 37 37
$$- \frac{\sqrt{37} \log{\left(- \frac{1}{6} + \frac{\sqrt{37}}{6} \right)}}{37} + \frac{\sqrt{37} \log{\left(- \frac{1}{6} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{37}}{6} \right)}}{37} - \frac{\sqrt{37} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{1}{6} + \frac{\sqrt{37}}{6} \right)} + i \pi\right)}{37} + \frac{\sqrt{37} \left(\log{\left(\frac{1}{6} + \frac{\sqrt{37}}{6} \right)} + i \pi\right)}{37}$$
-sqrt(37)*(pi*i + log(1/6 - tan(1/2) + sqrt(37)/6))/37 - sqrt(37)*log(-1/6 + sqrt(37)/6)/37 + sqrt(37)*(pi*i + log(1/6 + sqrt(37)/6))/37 + sqrt(37)*log(-1/6 + sqrt(37)/6 + tan(1/2))/37
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.