Sr Examen
Integral de 1/(1+3sinx+4cosx) dx
Solución
Ha introducido
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1 / | | 1 | ----------------------- dx | 1 + 3*sin(x) + 4*cos(x) | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(3 \sin{\left(x \right)} + 1\right) + 4 \cos{\left(x \right)}}\, dx$$
Integral(1/(1 + 3*sin(x) + 4*cos(x)), (x, 0, 1))
Respuesta (Indefinida)
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/ ___ \ / ___ \ / ___ | 2*\/ 6 /x\| ___ | 2*\/ 6 /x\| | \/ 6 *log|-1 - ------- + tan|-|| \/ 6 *log|-1 + ------- + tan|-|| | 1 \ 3 \2// \ 3 \2// | ----------------------- dx = C - -------------------------------- + -------------------------------- | 1 + 3*sin(x) + 4*cos(x) 12 12 | /
$$\int \frac{1}{\left(3 \sin{\left(x \right)} + 1\right) + 4 \cos{\left(x \right)}}\, dx = C + \frac{\sqrt{6} \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 + \frac{2 \sqrt{6}}{3} \right)}}{12} - \frac{\sqrt{6} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{2 \sqrt{6}}{3} - 1 \right)}}{12}$$
Gráfica
Respuesta
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/ / ___\\ / ___\ / / ___\\ / ___ \
___ | | 2*\/ 6 || ___ | 2*\/ 6 | ___ | | 2*\/ 6 || ___ | 2*\/ 6 |
\/ 6 *|pi*I + log|1 - tan(1/2) + -------|| \/ 6 *log|-1 + -------| \/ 6 *|pi*I + log|1 + -------|| \/ 6 *log|-1 + ------- + tan(1/2)|
\ \ 3 // \ 3 / \ \ 3 // \ 3 /
- ------------------------------------------ - ----------------------- + ------------------------------- + ----------------------------------
12 12 12 12
$$\frac{\sqrt{6} \log{\left(-1 + \tan{\left(\frac{1}{2} \right)} + \frac{2 \sqrt{6}}{3} \right)}}{12} - \frac{\sqrt{6} \log{\left(-1 + \frac{2 \sqrt{6}}{3} \right)}}{12} - \frac{\sqrt{6} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 1 + \frac{2 \sqrt{6}}{3} \right)} + i \pi\right)}{12} + \frac{\sqrt{6} \left(\log{\left(1 + \frac{2 \sqrt{6}}{3} \right)} + i \pi\right)}{12}$$
=
=
/ / ___\\ / ___\ / / ___\\ / ___ \
___ | | 2*\/ 6 || ___ | 2*\/ 6 | ___ | | 2*\/ 6 || ___ | 2*\/ 6 |
\/ 6 *|pi*I + log|1 - tan(1/2) + -------|| \/ 6 *log|-1 + -------| \/ 6 *|pi*I + log|1 + -------|| \/ 6 *log|-1 + ------- + tan(1/2)|
\ \ 3 // \ 3 / \ \ 3 // \ 3 /
- ------------------------------------------ - ----------------------- + ------------------------------- + ----------------------------------
12 12 12 12
$$\frac{\sqrt{6} \log{\left(-1 + \tan{\left(\frac{1}{2} \right)} + \frac{2 \sqrt{6}}{3} \right)}}{12} - \frac{\sqrt{6} \log{\left(-1 + \frac{2 \sqrt{6}}{3} \right)}}{12} - \frac{\sqrt{6} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 1 + \frac{2 \sqrt{6}}{3} \right)} + i \pi\right)}{12} + \frac{\sqrt{6} \left(\log{\left(1 + \frac{2 \sqrt{6}}{3} \right)} + i \pi\right)}{12}$$
-sqrt(6)*(pi*i + log(1 - tan(1/2) + 2*sqrt(6)/3))/12 - sqrt(6)*log(-1 + 2*sqrt(6)/3)/12 + sqrt(6)*(pi*i + log(1 + 2*sqrt(6)/3))/12 + sqrt(6)*log(-1 + 2*sqrt(6)/3 + tan(1/2))/12
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.