Sr Examen
Integral de dx/(3-2*cos(x)^2) dx
Solución
Ha introducido
[src]
pi / | | 1 | ------------- dx | 2 | 3 - 2*cos (x) | / 0
$$\int\limits_{0}^{\pi} \frac{1}{3 - 2 \cos^{2}{\left(x \right)}}\, dx$$
Integral(1/(3 - 2*cos(x)^2), (x, 0, pi))
Respuesta (Indefinida)
[src]
/ /x pi\ / /x\ \\ / /x pi\ / /x\ \\ / /x pi\ / /x\ \\ / /x pi\ / /x\ \\
| |- - --| | tan|-| || | |- - --| | tan|-| || _____________ _____________ | |- - --| | tan|-| || _____________ _____________ | |- - --| | tan|-| ||
| |2 2 | | \2/ || ___ | |2 2 | | \2/ || / ___ / ___ | |2 2 | | \2/ || ___ / ___ / ___ | |2 2 | | \2/ ||
5*|pi*floor|------| + atan|----------------|| 2*\/ 6 *|pi*floor|------| + atan|----------------|| 5*\/ 5 - 2*\/ 6 *\/ 5 + 2*\/ 6 *|pi*floor|------| + atan|----------------|| 2*\/ 6 *\/ 5 - 2*\/ 6 *\/ 5 + 2*\/ 6 *|pi*floor|------| + atan|----------------||
/ | \ pi / | _____________|| | \ pi / | _____________|| | \ pi / | _____________|| | \ pi / | _____________||
| | | / ___ || | | / ___ || | | / ___ || | | / ___ ||
| 1 \ \\/ 5 - 2*\/ 6 // \ \\/ 5 - 2*\/ 6 // \ \\/ 5 + 2*\/ 6 // \ \\/ 5 + 2*\/ 6 //
| ------------- dx = C + ----------------------------------------------- + --------------------------------------------------- + ------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------
| 2 _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________
| 3 - 2*cos (x) / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
| 27*\/ 5 - 2*\/ 6 + 11*\/ 6 *\/ 5 - 2*\/ 6 27*\/ 5 - 2*\/ 6 + 11*\/ 6 *\/ 5 - 2*\/ 6 27*\/ 5 - 2*\/ 6 + 11*\/ 6 *\/ 5 - 2*\/ 6 27*\/ 5 - 2*\/ 6 + 11*\/ 6 *\/ 5 - 2*\/ 6
/
$$\int \frac{1}{3 - 2 \cos^{2}{\left(x \right)}}\, dx = C + \frac{2 \sqrt{6} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{5 - 2 \sqrt{6}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{11 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{5 - 2 \sqrt{6}}} + \frac{5 \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{5 - 2 \sqrt{6}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{11 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{5 - 2 \sqrt{6}}} + \frac{2 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} \sqrt{2 \sqrt{6} + 5} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{2 \sqrt{6} + 5}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{11 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{5 - 2 \sqrt{6}}} + \frac{5 \sqrt{5 - 2 \sqrt{6}} \sqrt{2 \sqrt{6} + 5} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{2 \sqrt{6} + 5}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{11 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{5 - 2 \sqrt{6}}}$$
Gráfica
Respuesta
[src]
_____________ _____________ _____________ _____________
___ ___ / ___ / ___ ___ / ___ / ___
5*pi + 2*pi*\/ 6 5*pi 2*pi*\/ 6 5*pi*\/ 5 - 2*\/ 6 *\/ 5 + 2*\/ 6 2*pi*\/ 6 *\/ 5 - 2*\/ 6 *\/ 5 + 2*\/ 6
- ----------------------------------------------- + ----------------------------------------------- + ----------------------------------------------- + ----------------------------------------------- + -----------------------------------------------
_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________
/ ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
27*\/ 5 - 2*\/ 6 + 11*\/ 6 *\/ 5 - 2*\/ 6 27*\/ 5 - 2*\/ 6 + 11*\/ 6 *\/ 5 - 2*\/ 6 27*\/ 5 - 2*\/ 6 + 11*\/ 6 *\/ 5 - 2*\/ 6 27*\/ 5 - 2*\/ 6 + 11*\/ 6 *\/ 5 - 2*\/ 6 27*\/ 5 - 2*\/ 6 + 11*\/ 6 *\/ 5 - 2*\/ 6
$$- \frac{2 \sqrt{6} \pi + 5 \pi}{11 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{5 - 2 \sqrt{6}}} + \frac{2 \sqrt{6} \pi}{11 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{5 - 2 \sqrt{6}}} + \frac{2 \sqrt{6} \pi \sqrt{5 - 2 \sqrt{6}} \sqrt{2 \sqrt{6} + 5}}{11 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{5 - 2 \sqrt{6}}} + \frac{5 \pi}{11 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{5 - 2 \sqrt{6}}} + \frac{5 \pi \sqrt{5 - 2 \sqrt{6}} \sqrt{2 \sqrt{6} + 5}}{11 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{5 - 2 \sqrt{6}}}$$
=
=
_____________ _____________ _____________ _____________
___ ___ / ___ / ___ ___ / ___ / ___
5*pi + 2*pi*\/ 6 5*pi 2*pi*\/ 6 5*pi*\/ 5 - 2*\/ 6 *\/ 5 + 2*\/ 6 2*pi*\/ 6 *\/ 5 - 2*\/ 6 *\/ 5 + 2*\/ 6
- ----------------------------------------------- + ----------------------------------------------- + ----------------------------------------------- + ----------------------------------------------- + -----------------------------------------------
_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________
/ ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
27*\/ 5 - 2*\/ 6 + 11*\/ 6 *\/ 5 - 2*\/ 6 27*\/ 5 - 2*\/ 6 + 11*\/ 6 *\/ 5 - 2*\/ 6 27*\/ 5 - 2*\/ 6 + 11*\/ 6 *\/ 5 - 2*\/ 6 27*\/ 5 - 2*\/ 6 + 11*\/ 6 *\/ 5 - 2*\/ 6 27*\/ 5 - 2*\/ 6 + 11*\/ 6 *\/ 5 - 2*\/ 6
$$- \frac{2 \sqrt{6} \pi + 5 \pi}{11 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{5 - 2 \sqrt{6}}} + \frac{2 \sqrt{6} \pi}{11 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{5 - 2 \sqrt{6}}} + \frac{2 \sqrt{6} \pi \sqrt{5 - 2 \sqrt{6}} \sqrt{2 \sqrt{6} + 5}}{11 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{5 - 2 \sqrt{6}}} + \frac{5 \pi}{11 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{5 - 2 \sqrt{6}}} + \frac{5 \pi \sqrt{5 - 2 \sqrt{6}} \sqrt{2 \sqrt{6} + 5}}{11 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{5 - 2 \sqrt{6}}}$$
-(5*pi + 2*pi*sqrt(6))/(27*sqrt(5 - 2*sqrt(6)) + 11*sqrt(6)*sqrt(5 - 2*sqrt(6))) + 5*pi/(27*sqrt(5 - 2*sqrt(6)) + 11*sqrt(6)*sqrt(5 - 2*sqrt(6))) + 2*pi*sqrt(6)/(27*sqrt(5 - 2*sqrt(6)) + 11*sqrt(6)*sqrt(5 - 2*sqrt(6))) + 5*pi*sqrt(5 - 2*sqrt(6))*sqrt(5 + 2*sqrt(6))/(27*sqrt(5 - 2*sqrt(6)) + 11*sqrt(6)*sqrt(5 - 2*sqrt(6))) + 2*pi*sqrt(6)*sqrt(5 - 2*sqrt(6))*sqrt(5 + 2*sqrt(6))/(27*sqrt(5 - 2*sqrt(6)) + 11*sqrt(6)*sqrt(5 - 2*sqrt(6)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.