Limits & Integrals

Keep in mind $\displaystyle\lim_{p\rightarrow \infty}b^p = \begin{cases} 0 & b<1 \\ 1 & b=1 \\ \infty & b>1 \end{cases}$ Therefore, $f(x) = \begin{cases} \frac{x^2}{1+0} & \arctan{x}<1 \\ \frac{x^2}{1+1} & \arctan{x}=1 \\ \frac{x^2}{1+\infty} & \arctan{x}>1 \end{cases}$ Or, in simpler terms: $f(x) = \begin{cases} x^2 & x<\tan(1) \\ \frac{x^2}{2} & x=\tan(1) \\ 0 & x>\tan(1) \end{cases}$

Because it is piecewise, we can integrate $f$ in 3 separate parts:

$\displaystyle\int_{0}^{\tan(1)}x^2 dx+\displaystyle\int_{\tan(1)}^{\tan(1)}\frac{x^2}{2}dx+\displaystyle\int_{\tan(1)}^{\infty} 0~dx$

$=\frac{\tan^3(1)}{3}-0+0+0\approx 1.259$ $\beta_{\lceil \mid \rceil}$