Daily math problems for June 30, 2026 — Logs, Trigonometry, Integrals

$\bullet$ Invert $y=3{\log }_{2}x$:

$${\log }_{2}x=\frac{y}{3}\Rightarrow x=2^{y\mathrm{/}3}$$

$\bullet$ So $f^{-1}(y)=2^{y\mathrm{/}3}$.

$\bullet$ Evaluate at $y=27$:

$$f^{-1}(27)=2^{27\mathrm{/}3}=2^9=512$$

$\bullet$ Check: $f(512)=3{\log }_{2}512=3\cdot 9=27$.

$\cos u=-\frac{1}{2}$ on $[0,4\pi )$ at $u=\frac{2\pi }{3},\frac{4\pi }{3},\frac{8\pi }{3},\frac{10\pi }{3}$. Dividing by $2$: $x=\frac{\pi }{3},\frac{2\pi }{3},\frac{4\pi }{3},\frac{5\pi }{3}$. Four solutions.

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}dx=\sqrt{\pi }$. By even symmetry, ${\int }_0^{\mathrm{\infty }}{e}^{-x^2}dx={\displaystyle \frac{\sqrt{\pi }}{2}}$.