Daily Math · 2026-05-24

Question 1

Calculați $\log_4 8$.

Accepted Answer

Folosind schimbarea de bază: $\log_4 8 = \dfrac{\log_2 8}{\log_2 4} = \dfrac{3}{2}$.

Answer

$\dfrac{1}{2}$

Answer

$\dfrac{2}{3}$

Answer

$\dfrac{3}{2}$

Answer

$2$

Question 2

Find $\displaystyle\int\dfrac{1}{x^2-1}\,dx$.

Accepted Answer

$\bullet$ Decompose by partial fractions:

$$\dfrac{1}{x^2-1}=\dfrac{1}{2}\left(\dfrac{1}{x-1}-\dfrac{1}{x+1}\right)$$

$\bullet$ Integrate term-by-term:

$$\int\dfrac{dx}{x^2-1}=\dfrac{1}{2}\bigl(\ln|x-1|-\ln|x+1|\bigr)+C$$

$\bullet$ Combine logs:

$$=\dfrac{1}{2}\ln\left|\dfrac{x-1}{x+1}\right|+C$$

Answer

$\ln\!\left|\dfrac{x-1}{x+1}\right| + C$

Answer

$\dfrac{1}{2}\ln\!\left|\dfrac{x-1}{x+1}\right| + C$

Answer

$\arctan x + C$

Answer

$\ln|x^2-1| + C$

Answer

$\dfrac{1}{2}\ln|x^2-1| + C$

Answer

$\dfrac{1}{2}\ln\!\left|\dfrac{x+1}{x-1}\right| + C$

Question 3

By the rational-root theorem, possible rational roots of $P(X) = 2X^3 + 3X^2 - 1$ are of the form $\dfrac{p}{q}$ with $p \mid 1$ and $q \mid 2$. They are:

Accepted Answer

$\dfrac{p}{q}$ runs over $\!\left\{\pm 1, \pm\dfrac{1}{2}\right\}$.

Answer

$\pm 1, \pm 2$

Answer

$\pm 1, \pm \dfrac{1}{2}$

Answer

$\pm 1, \pm 3, \pm 2$

Answer

$\pm 1$ only

Problems & worked solutions

Problem 1 — Logarithms

Problem 2 — Integrals

Problem 3 — Polynomial Rings