Enigma

¡Qué cosa con la integral definida!

Nos dispusimos a trabajar la siguiente integral definida:

[math] \int_0^{\infty } \frac{\sqrt{n}}{n^2+1} \, dn [/math]

Tiene sus dificultades pero después da hacer varias sustituciones, resolviendo las integrales racionales con raíces imaginarias, obtuvimos la siguiente primera solución:

[math] \frac{\sqrt{2}}{2}\left[\frac{1}{2}.\text{Log}\left[\frac{1+x-\sqrt{2x}}{1+x+\sqrt{2x}}\right]+\text{ArcTan}\left[\sqrt{2x}-1\right]+\text{ArcTan}\left[\sqrt{2x}+1\right]\right]_0^{\infty } [/math]

Una vez aplicado el límite con "n" tendiendo a infinito y reemplazando por "n=0" llegamos a la respuesta casi definitiva:

[math] \frac{\sqrt{2}}{2}\left[\frac{1}{2}.0+\frac{\pi }{2}+\frac{\pi }{2}\right]-\frac{\sqrt{2}}{2}\left[\frac{1}{2}.0+\text{ArcTan}[-1]+\text{ArcTan}[1]\right] [/math]

Lo curioso y he aquí el enigma fue:

  1. Si consideramos que [math] \text{arctg} (-1)= \frac{3}{4}\pi [/math] , por ser del segundo cuadrante, y [math] \text{arctg} (1)= \frac{1}{4}\pi [/math] , el cálculo definitivo es el siguiente:

  2. [math] \frac{\sqrt{2}}{2}.\pi -\frac{\sqrt{2}}{2}\left(\frac{3}{4}\pi +\frac{1}{4}\pi \right)=\frac{\sqrt{2}}{2}(\pi -\pi )=0 [/math]

  3. Si consideramos que [math] \text{arctg} (-1)= \frac{-1}{4}\pi [/math], por ser del cuarto cuadrante, y [math] \text{arctg} (1)= \frac{1}{4}\pi [/math], el cálculo definitivo es el siguiente:

  4. [math] \frac{\sqrt{2}}{2}.\pi -\frac{\sqrt{2}}{2}\left(-\frac{1}{4}\pi +\frac{1}{4}\pi \right)=\frac{\sqrt{2}}{2}\pi [/math]

  5. Si consideramos que [math] \text{arctg} (-1)= \frac{7}{4}\pi [/math] , por ser del cuarto cuadrante, y [math] \text{arctg} (1)= \frac{1}{4}\pi [/math] , el cálculo definitivo es el siguiente:

  6. [math] \frac{\sqrt{2}}{2}.\pi -\frac{\sqrt{2}}{2}\left(\frac{7}{4}\pi +\frac{1}{4}\pi \right)=\frac{\sqrt{2}}{2}\pi (\pi -2\pi )=-\frac{\sqrt{2}}{2}\pi [/math]

Por consiguiente la integral definida planteada tiene muchas soluciones.

  • Este es el momento a analizar en conjunto el enigma aquí planteado.

  • Si sos estudiante de una Universidad, te pondrás a analizar y discutir con tus compañeros.

  • Si usted es docente, ¿Qué mejor que este enigma para trabajar en clase, verdad?

5
Su voto: Nada Promedio: 5 (3 votos)

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lo primero que hay que hacer,

lo primero que hay que hacer, siempre, para no cometer errores es "leer correctamente el anunciado y saber de qué se trata y con qué contamos"

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