Indefinite Integration
Logarithmic Forms
Grade 12
Question:
<p><span>\(\int \frac{g(x)}{f(x)}\)</span> dx is equal to</p>
<p>(A) <span>\(\ln(e^x + e^{-x}) + C\)</span></p>
<p>(B) <span>\(\ln(e^x - e^{-x}) + C\)</span></p>
<p>(C) <span>\(\ln(e^{2x} - 1) + C\)</span></p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: Recognize the structure of exponential integrals and use hyperbolic function properties.
<p>Using the matrix exponential definition and properties of the given matrix <span>$A = \begin{pmatrix} x & x \\ -x & -x \end{pmatrix}$</span>, we evaluate <span>$f(x)$</span> and <span>$g(x)$</span> from <span>$e^A$</span>, then compute the integral.</p>
Correct Answer: A