Sr Examen
Integral de cos(x)/(cos^2(x)-5*cos(x)+6) dx
Solución
Ha introducido
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1 / | | cos(x) | ---------------------- dx | 2 | cos (x) - 5*cos(x) + 6 | / 0
$$\int\limits_{0}^{1} \frac{\cos{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} - 5 \cos{\left(x \right)}\right) + 6}\, dx$$
Integral(cos(x)/(cos(x)^2 - 5*cos(x) + 6), (x, 0, 1))
Respuesta (Indefinida)
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/ /x pi\ \ / /x pi\ \
| |- - --| | | |- - --| |
/ ___ | |2 2 | / ___ /x\\| ___ | |2 2 | / ___ /x\\|
| 3*\/ 2 *|pi*floor|------| + atan|\/ 2 *tan|-||| 4*\/ 3 *|pi*floor|------| + atan|\/ 3 *tan|-|||
| cos(x) \ \ pi / \ \2/// \ \ pi / \ \2///
| ---------------------- dx = C - ----------------------------------------------- + -----------------------------------------------
| 2 2 3
| cos (x) - 5*cos(x) + 6
|
/
$$\int \frac{\cos{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} - 5 \cos{\left(x \right)}\right) + 6}\, dx = C - \frac{3 \sqrt{2} \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2} + \frac{4 \sqrt{3} \left(\operatorname{atan}{\left(\sqrt{3} \tan{\left(\frac{x}{2} \right)} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{3}$$
Gráfica
Respuesta
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___ ___ / / ___ \\ ___ ___ / / ___ \\
3*pi*\/ 2 3*\/ 2 *\-pi + atan\\/ 2 *tan(1/2)// 4*pi*\/ 3 4*\/ 3 *\-pi + atan\\/ 3 *tan(1/2)//
- ---------- - ------------------------------------ + ---------- + ------------------------------------
2 2 3 3
$$- \frac{3 \sqrt{2} \pi}{2} + \frac{4 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\sqrt{3} \tan{\left(\frac{1}{2} \right)} \right)}\right)}{3} - \frac{3 \sqrt{2} \left(- \pi + \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{1}{2} \right)} \right)}\right)}{2} + \frac{4 \sqrt{3} \pi}{3}$$
=
=
___ ___ / / ___ \\ ___ ___ / / ___ \\
3*pi*\/ 2 3*\/ 2 *\-pi + atan\\/ 2 *tan(1/2)// 4*pi*\/ 3 4*\/ 3 *\-pi + atan\\/ 3 *tan(1/2)//
- ---------- - ------------------------------------ + ---------- + ------------------------------------
2 2 3 3
$$- \frac{3 \sqrt{2} \pi}{2} + \frac{4 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\sqrt{3} \tan{\left(\frac{1}{2} \right)} \right)}\right)}{3} - \frac{3 \sqrt{2} \left(- \pi + \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{1}{2} \right)} \right)}\right)}{2} + \frac{4 \sqrt{3} \pi}{3}$$
-3*pi*sqrt(2)/2 - 3*sqrt(2)*(-pi + atan(sqrt(2)*tan(1/2)))/2 + 4*pi*sqrt(3)/3 + 4*sqrt(3)*(-pi + atan(sqrt(3)*tan(1/2)))/3
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.