Indefinite Integration
Substitution Method
Grade None
Question:
<p><span>\(\int \frac{\ln 2x}{4x}\)</span> dx is equal to</p>
<p>(A) <span>\(\frac{(\ln 2x)^2}{2} + C\)</span></p>
<p>(B) <span>\(\frac{(\ln 2x)^2}{2} + C\)</span></p>
<p>(C) <span>\(\frac{(\ln 2x+1)^2}{2} + C\)</span></p>
<p>(D) <span>\(\frac{(\ln 2x)^2}{2} + C\)</span></p>
Step-by-Step Solution
Key Concept: Use substitution with the logarithmic function as the substitution variable.
<p>Let <span>$u = \ln 2x$</span>, then <span>$du = \frac{1}{x}dx$</span>. The integral becomes <span>$\int \frac{u}{4} du = \frac{u^2}{8} + C = \frac{(\ln 2x)^2}{8} + C$</span>. However, we need to verify the exact form based on the integration.</p>
Correct Answer: A