Indefinite Integration
Integration of trigonometric functions
Grade 12
Question:
<p>Let \(a \in (0, \pi/2)\) be fixed. If the integral \(\int \frac{\tan x - \tan a}{\tan x + \tan a} dx = A(x) \cos 2a + B(x) \sin 2a + C\), where \(C\) is a constant of integration, then the functions \(A(x)\) and \(B(x)\) are respectively</p>
<p>(a) \(x + a\) and \(\log_e|\sin(x+a)|\)</p>
<p>(b) \(x - a\) and \(\log_e|\sin(x-a)|\)</p>
<p>(c) \(x - a\) and \(\log_e|\cos(x-a)|\)</p>
<p>(d) \(x + a\) and \(\log_e|\sin(x-a)|\)</p>
Step-by-Step Solution
Key Concept: Use tangent difference formula and convert to simpler trigonometric forms for integration.
<p><strong>Solution:</strong> Using the identity $\frac{\tan x - \tan a}{\tan x + \tan a} = \frac{\sin(x-a)}{\cos(x-a)\cos(x+a)}$</p><p>After integration and simplification using angle formulas and trigonometric identities: $A(x) = x - a$ and $B(x) = \log_e|\sin(x-a)|$</p>
Correct Answer: B