Indefinite Integration
Integration of Trigonometric Functions
Grade 12
Question:
<p>If <span>\(\int \frac{dx}{(\sin x + 4)(\sin x - 1)} = A \tan^{-1}\left(\frac{\tan x/2 - 1}{2}\right) + B \tan^{-1} f(x) + C\)</span>, then find the value of A.</p>
<p>(A) <span>\(A = \frac{1}{4}\)</span></p>
<p>(B) <span>\(A = \frac{1}{5}\)</span></p>
<p>(C) Other value</p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: Use Weierstrass substitution to convert trigonometric integrals into rational form, then use partial fractions.
<p><strong>Step 1:</strong> Use Weierstrass substitution <span>$t = \tan(x/2)$</span> to convert to rational form.<br/><strong>Step 2:</strong> Apply partial fractions to decompose.<br/><strong>Step 3:</strong> Integrate each term to find the coefficient A.</p>
Correct Answer: A