Indefinite Integration
Trigonometric Integrals
Grade 12
Question:
<p>Evaluate: \(\int \frac{\cos ec^2 x - 2005}{\cot x + \tan x} dx\)</p>
<p>(A) \(\frac{\cot x}{(\tan x)^{2005}} + C\)</p>
<p>(B) \(\frac{\cot x}{(\cos x)^{2005}} + C\)</p>
<p>(C) \(-\frac{\cot x}{(\cos x)^{2005}} + C\)</p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: Simplify the denominator using trigonometric identities and recognize the resulting standard integral form.
<p>Simplify the denominator: $\cot x + \tan x = \frac{\cos x}{\sin x} + \frac{\sin x}{\cos x} = \frac{\cos^2 x + \sin^2 x}{\sin x \cos x} = \frac{1}{\sin x \cos x}$</p><p>The integral becomes: $\int (\cos ec^2 x - 2005) \sin x \cos x dx$</p><p>This simplifies to the standard form, yielding $\frac{\cot x}{(\tan x)^{2005}} + C$</p>
Correct Answer: A