Definite Integration
Trigonometric Integrals
Grade 12
Question:
<p>The value of the definite integral \[\int_0^{\pi/2} \frac{dx}{\tan x + \cot x + \csc x + \sec x}\]</p>
<p>(a) \(1 - \frac{\pi}{4}\)</p>
<p>(b) \(\frac{\pi}{4} + 1\)</p>
<p>(c) \(\pi + \frac{1}{4}\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Convert all trigonometric functions to sine and cosine, then simplify the denominator.
<p><strong>Solution:</strong> Rewrite the denominator in terms of sine and cosine:</p><p>$$\tan x + \cot x + \csc x + \sec x = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} + \frac{1}{\sin x} + \frac{1}{\cos x}$$</p><p>$$= \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} + \frac{\sin x + \cos x}{\sin x \cos x} = \frac{1 + \sin x + \cos x}{\sin x \cos x}$$</p><p>The integral becomes tractable via substitution or residue methods, yielding $1 - \frac{\pi}{4}$.</p>
Correct Answer: a