Daily math problems for May 24, 2026 — Logaritmi, Integrals, Algebra

Folosind schimbarea de bază: ${\log }_{4}8={\displaystyle \frac{{\log }_{2}8}{{\log }_{2}4}}={\displaystyle \frac{3}{2}}$.

$\bullet$ Decompose by partial fractions:

$${\displaystyle \frac{1}{x^2-1}}={\displaystyle \frac{1}{2}}({\displaystyle \frac{1}{x-1}}-{\displaystyle \frac{1}{x+1}})$$

$\bullet$ Integrate term-by-term:

$$\int {\displaystyle \frac{dx}{x^2-1}}={\displaystyle \frac{1}{2}}(\ln \mathrm{\mid }x-1\mathrm{\mid }-\ln \mathrm{\mid }x+1\mathrm{\mid })+C$$

$\bullet$ Combine logs:

$$={\displaystyle \frac{1}{2}}\ln \mid {\displaystyle \frac{x-1}{x+1}}\mid +C$$

${\displaystyle \frac{p}{q}}$ runs over $\text{\hspace{0.17em}⁣}\{\pm 1,\pm {\displaystyle \frac{1}{2}}\}$.