Indefinite Integration
Integration of Exponential and Trigonometric Functions
Grade 12
Question:
<p>Evaluate: <equation>\int \frac{\cos 2x - 2x \operatorname{cosec}^2 2005}{\sin^2 x \cos x} dx</equation></p>
<p>(a) <equation>-\frac{1}{a} e^{2ax} \cos\left(\frac{\pi}{4} + ax\right) + C</equation></p>
<p>(b) <equation>-\frac{1}{2a} e^{2ax} \cot\left(\frac{\pi}{4} + ax\right) + C</equation></p>
<p>(c) <equation>-\frac{1}{2a} e^{2ax} \cos\left(\frac{\pi}{4} + ax\right) + C</equation></p>
<p>(d) <equation>-\frac{1}{a} e^{2ax} \operatorname{cosec}\left(\frac{\pi}{4} + ax\right) + C</equation></p>
Step-by-Step Solution
Key Concept: Convert complex trigonometric expressions to standard forms and use exponential-trigonometric integration rules.
<p><strong>Step 1:</strong> Simplify the integrand using trigonometric identities.</p><p><strong>Step 2:</strong> Use substitution to evaluate the exponential-trigonometric integral.</p><p>∴ Answer is (b).</p>
Correct Answer: B