Indefinite Integration
Integration of Trigonometric Functions
Grade 12
Question:
<p>If <span>\(\int \frac{1+3\tan x(\tan x + \sec x)}{\tan x} dx = a \log \left|\cos \frac{x}{2} + \sin \frac{x}{2}\right| + C\)</span> where <span>\(0 < x < \pi\)</span>, then <span>\(a\)</span> is equal to</p>
<p>(P) 0</p>
<p>(Q) –2</p>
<p>(R) 4</p>
Step-by-Step Solution
Key Concept: Decompose the complex trigonometric integrand into simpler parts and use standard integral formulas for trigonometric functions.
<p><strong>Step 1:</strong> Expand the integrand: <span>$\int \frac{1+3\tan^2 x + 3\tan x \sec x}{\tan x} dx$</span></p><p><strong>Step 2:</strong> Separate into parts: <span>$\int \frac{1}{\tan x} dx + \int 3\tan x dx + \int 3\sec x dx$</span></p><p><strong>Step 3:</strong> Evaluate each integral using standard formulas</p><p><strong>Step 4:</strong> Simplify and match with the given form <span>$a \log \left|\cos \frac{x}{2} + \sin \frac{x}{2}\right| + C$</span></p><p><strong>Step 5:</strong> By comparing coefficients, <span>$a = -2$</span></p><p>∴ Answer is Q.</p>
Correct Answer: Q