Indefinite Integration
Trigonometric Integrals
Grade 12
Question:
<p>Evaluate \(\int \frac{(\sin x - 2\cos x)}{(2\sin x + \cos x)} dx\)</p>
<p>(a) \(\frac{\tan x - 2}{2\tan x + 1} + C\)</p>
<p>(b) \(\frac{1}{4} \log\left|\frac{\tan x - 2}{\tan x + 2}\right| + C\)</p>
<p>(c) \(\frac{1}{4} \log\left|\frac{2\sin x + \cos x}{\sin x + 2\cos x}\right| + C\)</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: Decompose the numerator in terms of the denominator and its derivative to create standard logarithmic integral forms.
<p>Use the decomposition method where the numerator is expressed as $A(2\sin x + \cos x) + B(2\cos x - \sin x)$. Solving for constants and integrating yields $\frac{1}{4} \log\left|\frac{2\sin x + \cos x}{\sin x + 2\cos x}\right| + C$.</p>
Correct Answer: C