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