Find the Integral $\int \frac{d x}{x(x-2)}$ A) $2 \ln \left|\frac{x^{2}-2}{x}\right|+C$ B) $2 \ln \left|\frac{x-2}{x}\right|+C$

question 23

Multiple Choice

Find the integral. $\int \frac{d x}{x(x-2) }$

• A) \(2 \ln \left|\frac{x^{2}-2}{x}\right|+C\)
• B) \(2 \ln \left|\frac{x-2}{x}\right|+C\)
• C) \(\frac{1}{2} \ln \left|\frac{x^{2}-2}{x}\right|+C\)
• D) \(\frac{1}{2} \ln \left|\frac{x-2}{x}\right|+C\)

---

Definitions:

Acidity

A measure of the concentration of hydrogen ions in a solution, often associated with the degree to which a substance is able to donate a proton.

Phenol

An aromatic compound consisting of a benzene ring bonded to a hydroxyl group (-OH).

Ethanol

A volatile, flammable, clear liquid also known as alcohol, used as a solvent, in the manufacture of various chemicals, and as an intoxicant.

Acetic Acid

A weak carboxylic acid with the formula CH3COOH, known for its sour taste and pungent smell, used in vinegar.