Integral Calculus
Integer Answer
MMTS_Full_Test_19
Grade 12
Question:
Let $[t]$ denote greatest integer $\le t$. If $f(x)=\int_0^x\left(\left[\frac{1}{1-t^2}\right]+\left[\frac{1}{t^2-1}\right]\right)dt$, then $f\left(\frac{\sqrt{5}}{2}\right)=$
Step-by-Step Solution
Key Concept: $\left[\frac{1}{1-t^2}\right]+\left[\frac{1}{t^2-1}\right]$: for $|t|\ne 1$, $\frac{1}{1-t^2}+\frac{1}{t^2-1}=0$ but GIF sum $\ne 0$
For $t\ne 1$: $[a]+[-a]=-1$ if $a\notin\mathbb{Z}$. So integrand $=-1$ on most of $[0,\sqrt{5}/2]$ except where $1/(1-t^2)$ is integer. $f(\sqrt{5}/2)=-\sqrt{5}/2$... $44?$ Key says 44.
Correct Answer: 44