Nice Function

Calculus Level 5

Let $\displaystyle I(b)=\lim _{ n\rightarrow \infty }{ \int _{ b }^{ \infty }{ \frac { n }{ 1+{ n }^{ 2 }{ x }^{ 2 } } dx } }$ , where $b$ is a real number.

Let the sum of all the values $I(x)$ can take be $A$ . Find $\lfloor A \rfloor$ .

The answer is 4.

1 solution

Nishant Sharma
Jul 22, 2014

Pretty easy. The integral is straightforward and evaluates to $\displaystyle\frac{\pi}{2}-\tan^{-1}(bn)$ . Now since $b\in\mathbb{R}$ so let's take three simple cases:

CASE I: $b<0$ Now applying limit we get $I=\pi$

CASE II: $b=0$ This gives $\displaystyle\,I=\frac{\pi}{2}$

CASE III: $b>0$ This gives $I=0$

Now summing them up we have $\displaystyle\lfloor\,A\rfloor=\lfloor\frac{3\pi}{2}\rfloor=\boxed{4}$ .

Thanks, I have updated the question accordingly.

Calvin Lin Staff - 6 years, 10 months ago

Nice Function

The answer is 4.

1 solution

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