Indefinite Integration
Integration of Trigonometric Functions
Grade 12
Question:
<p>Integral of \(\sqrt{1+2\cot x(\cot x+\csc x)}\) w.r.t. \(x\) is</p>
<li>\(2\ln\!\left|\sin\dfrac{x}{2}\right|+C\)</li>
<li>\(2\ln\!\left|\cos\dfrac{x}{2}\right|+C\)</li>
<li>\(\ln|\sin x|+C\)</li>
<li>\(\ln|\csc x-\cot x|+C\)</li>
Step-by-Step Solution
Key Concept: Simplify under the radical: 1+2cot^2x+2cotx \cdot cscx = (cscx+cotx)^2. Then integrate.
<p><strong>Simplification:</strong></p>
<p>\[1+2\cot x(\cot x+\csc x) = 1+2\cot^2 x+2\cot x\csc x\]</p>
<p>\[= \underbrace{(1+\cot^2 x)}_{\csc^2 x} + \cot^2 x + 2\cot x\csc x = (\csc x+\cot x)^2\]</p>
<p>So the integral becomes \(\displaystyle\int(\csc x+\cot x)\,dx\) (taking positive root).</p>
<p>\[= \ln|\csc x-\cot x|+\ln|\sin x|+C = \ln\!\left|\frac{\sin x}{\csc x-\cot x}\right|+C\]</p>
<p>Since \(\csc x-\cot x = \tan(x/2)\):</p>
<p>\[\int(\csc x+\cot x)\,dx = -\ln|\csc x+\cot x|+\ln|\sin x|+C = 2\ln|\sin(x/2)|+C\]</p>
<p>Answer: <strong>(A)</strong></p>
Correct Answer: A