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