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