Sr Examen
Integral de 1/(sin^2x+2cos^2x) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------------- dx | 2 2 | sin (x) + 2*cos (x) | / 0
$$\int\limits_{0}^{1} \frac{1}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}\, dx$$
Integral(1/(sin(x)^2 + 2*cos(x)^2), (x, 0, 1))
Respuesta (Indefinida)
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/ /x pi\ \ / /x pi\ \
| |- - --| | | |- - --| |
/ ___ | |2 2 | / ___ /x\\| ___ | |2 2 | / ___ /x\\|
| \/ 2 *|pi*floor|------| + atan|1 + \/ 2 *tan|-||| \/ 2 *|pi*floor|------| + atan|-1 + \/ 2 *tan|-|||
| 1 \ \ pi / \ \2/// \ \ pi / \ \2///
| ------------------- dx = C + ------------------------------------------------- + --------------------------------------------------
| 2 2 2 2
| sin (x) + 2*cos (x)
|
/
$$\int \frac{1}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}\, dx = C + \frac{\sqrt{2} \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2} + \frac{\sqrt{2} \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2}$$
Gráfica
Respuesta
[src]
___ / / ___ \\ ___ / / ___ \\
___ \/ 2 *\-pi - atan\1 - \/ 2 *tan(1/2)// \/ 2 *\-pi + atan\1 + \/ 2 *tan(1/2)//
pi*\/ 2 + -------------------------------------- + --------------------------------------
2 2
$$\frac{\sqrt{2} \left(- \pi - \operatorname{atan}{\left(- \sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{2} + \frac{\sqrt{2} \left(- \pi + \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{2} + \sqrt{2} \pi$$
=
=
___ / / ___ \\ ___ / / ___ \\
___ \/ 2 *\-pi - atan\1 - \/ 2 *tan(1/2)// \/ 2 *\-pi + atan\1 + \/ 2 *tan(1/2)//
pi*\/ 2 + -------------------------------------- + --------------------------------------
2 2
$$\frac{\sqrt{2} \left(- \pi - \operatorname{atan}{\left(- \sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{2} + \frac{\sqrt{2} \left(- \pi + \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{2} + \sqrt{2} \pi$$
pi*sqrt(2) + sqrt(2)*(-pi - atan(1 - sqrt(2)*tan(1/2)))/2 + sqrt(2)*(-pi + atan(1 + sqrt(2)*tan(1/2)))/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.