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