Sr Examen
Integral de 1/(1+4(cos(x))^2) dx
Solución
Ha introducido
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0 / | | 1 | ------------- dx | 2 | 1 + 4*cos (x) | / pi -- 4
$$\int\limits_{\frac{\pi}{4}}^{0} \frac{1}{4 \cos^{2}{\left(x \right)} + 1}\, dx$$
Integral(1/(1 + 4*cos(x)^2), (x, pi/4, 0))
Respuesta (Indefinida)
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/ /x pi\ \ / /x pi\ \
| |- - --| | | |- - --| |
/ ___ | |2 2 | / ___ /x\\| ___ | |2 2 | / ___ /x\\|
| \/ 5 *|pi*floor|------| + atan|-2 + \/ 5 *tan|-||| \/ 5 *|pi*floor|------| + atan|2 + \/ 5 *tan|-|||
| 1 \ \ pi / \ \2/// \ \ pi / \ \2///
| ------------- dx = C + -------------------------------------------------- + -------------------------------------------------
| 2 5 5
| 1 + 4*cos (x)
|
/
$$\int \frac{1}{4 \cos^{2}{\left(x \right)} + 1}\, dx = C + \frac{\sqrt{5} \left(\operatorname{atan}{\left(\sqrt{5} \tan{\left(\frac{x}{2} \right)} - 2 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{5} + \frac{\sqrt{5} \left(\operatorname{atan}{\left(\sqrt{5} \tan{\left(\frac{x}{2} \right)} + 2 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{5}$$
Gráfica
Respuesta
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___ / / ___ / ___\\\ ___ / / ___ / ___\\\ ___ ___
\/ 5 *\-pi - atan\2 - \/ 5 *\-1 + \/ 2 /// \/ 5 *\-pi + atan\2 + \/ 5 *\-1 + \/ 2 /// \/ 5 *(-pi - atan(2)) \/ 5 *(-pi + atan(2))
- ------------------------------------------ - ------------------------------------------ + --------------------- + ---------------------
5 5 5 5
$$\frac{\sqrt{5} \left(- \pi - \operatorname{atan}{\left(2 \right)}\right)}{5} + \frac{\sqrt{5} \left(- \pi + \operatorname{atan}{\left(2 \right)}\right)}{5} - \frac{\sqrt{5} \left(- \pi + \operatorname{atan}{\left(\sqrt{5} \left(-1 + \sqrt{2}\right) + 2 \right)}\right)}{5} - \frac{\sqrt{5} \left(- \pi - \operatorname{atan}{\left(- \sqrt{5} \left(-1 + \sqrt{2}\right) + 2 \right)}\right)}{5}$$
=
=
___ / / ___ / ___\\\ ___ / / ___ / ___\\\ ___ ___
\/ 5 *\-pi - atan\2 - \/ 5 *\-1 + \/ 2 /// \/ 5 *\-pi + atan\2 + \/ 5 *\-1 + \/ 2 /// \/ 5 *(-pi - atan(2)) \/ 5 *(-pi + atan(2))
- ------------------------------------------ - ------------------------------------------ + --------------------- + ---------------------
5 5 5 5
$$\frac{\sqrt{5} \left(- \pi - \operatorname{atan}{\left(2 \right)}\right)}{5} + \frac{\sqrt{5} \left(- \pi + \operatorname{atan}{\left(2 \right)}\right)}{5} - \frac{\sqrt{5} \left(- \pi + \operatorname{atan}{\left(\sqrt{5} \left(-1 + \sqrt{2}\right) + 2 \right)}\right)}{5} - \frac{\sqrt{5} \left(- \pi - \operatorname{atan}{\left(- \sqrt{5} \left(-1 + \sqrt{2}\right) + 2 \right)}\right)}{5}$$
-sqrt(5)*(-pi - atan(2 - sqrt(5)*(-1 + sqrt(2))))/5 - sqrt(5)*(-pi + atan(2 + sqrt(5)*(-1 + sqrt(2))))/5 + sqrt(5)*(-pi - atan(2))/5 + sqrt(5)*(-pi + atan(2))/5
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.