Sr Examen
Integral de sin2(x)/(4+cos^2(x)) dx
Solución
Ha introducido
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1 / | | 2 | sin (x) | ----------- dx | 2 | 4 + cos (x) | / 0
$$\int\limits_{0}^{1} \frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} + 4}\, dx$$
Integral(sin(x)^2/(4 + cos(x)^2), (x, 0, 1))
Respuesta (Indefinida)
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/ /x pi\ / ___ /x\\\ / /x pi\ / ___ /x\\\ / | |- - --| | \/ 5 *tan|-||| | |- - --| | \/ 5 *tan|-||| | ___ | |2 2 | |1 \2/|| ___ | |2 2 | | 1 \2/|| | 2 \/ 5 *|pi*floor|------| + atan|- + ------------|| \/ 5 *|pi*floor|------| + atan|- - + ------------|| | sin (x) \ \ pi / \2 2 // \ \ pi / \ 2 2 // | ----------- dx = C - x + ------------------------------------------------- + --------------------------------------------------- | 2 2 2 | 4 + cos (x) | /
$$\int \frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} + 4}\, dx = C - x + \frac{\sqrt{5} \left(\operatorname{atan}{\left(\frac{\sqrt{5} \tan{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2} + \frac{\sqrt{5} \left(\operatorname{atan}{\left(\frac{\sqrt{5} \tan{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2}$$
Gráfica
Respuesta
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/ / ___ \\ / / ___ \\
___ | |1 \/ 5 *tan(1/2)|| ___ | |1 \/ 5 *tan(1/2)||
\/ 5 *|-pi - atan|- - --------------|| \/ 5 *|-pi + atan|- + --------------|| ___ ___
\ \2 2 // \ \2 2 // \/ 5 *(-pi - atan(1/2)) \/ 5 *(-pi + atan(1/2))
-1 + -------------------------------------- + -------------------------------------- - ----------------------- - -----------------------
2 2 2 2
$$\frac{\sqrt{5} \left(- \pi - \operatorname{atan}{\left(- \frac{\sqrt{5} \tan{\left(\frac{1}{2} \right)}}{2} + \frac{1}{2} \right)}\right)}{2} + \frac{\sqrt{5} \left(- \pi + \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5} \tan{\left(\frac{1}{2} \right)}}{2} \right)}\right)}{2} - 1 - \frac{\sqrt{5} \left(- \pi + \operatorname{atan}{\left(\frac{1}{2} \right)}\right)}{2} - \frac{\sqrt{5} \left(- \pi - \operatorname{atan}{\left(\frac{1}{2} \right)}\right)}{2}$$
=
=
/ / ___ \\ / / ___ \\
___ | |1 \/ 5 *tan(1/2)|| ___ | |1 \/ 5 *tan(1/2)||
\/ 5 *|-pi - atan|- - --------------|| \/ 5 *|-pi + atan|- + --------------|| ___ ___
\ \2 2 // \ \2 2 // \/ 5 *(-pi - atan(1/2)) \/ 5 *(-pi + atan(1/2))
-1 + -------------------------------------- + -------------------------------------- - ----------------------- - -----------------------
2 2 2 2
$$\frac{\sqrt{5} \left(- \pi - \operatorname{atan}{\left(- \frac{\sqrt{5} \tan{\left(\frac{1}{2} \right)}}{2} + \frac{1}{2} \right)}\right)}{2} + \frac{\sqrt{5} \left(- \pi + \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5} \tan{\left(\frac{1}{2} \right)}}{2} \right)}\right)}{2} - 1 - \frac{\sqrt{5} \left(- \pi + \operatorname{atan}{\left(\frac{1}{2} \right)}\right)}{2} - \frac{\sqrt{5} \left(- \pi - \operatorname{atan}{\left(\frac{1}{2} \right)}\right)}{2}$$
-1 + sqrt(5)*(-pi - atan(1/2 - sqrt(5)*tan(1/2)/2))/2 + sqrt(5)*(-pi + atan(1/2 + sqrt(5)*tan(1/2)/2))/2 - sqrt(5)*(-pi - atan(1/2))/2 - sqrt(5)*(-pi + atan(1/2))/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.