Indefinite Integration
Trigonometric Integrals
Grade 12
Question:
<p>\(\int \frac{\sin 2x}{\sin^4 x + \cos^4 x} dx = a\cot^{-1}(b\tan 2x) + c\), then</p>
<p>(A) \(a = 1, b = -1\)</p>
<p>(B) \(a = -1, b = 1\)</p>
<p>(C) \(a = -1, b = -1\)</p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: Simplify the denominator using trigonometric identities and recognize the resulting form as an inverse cotangent integral.
<p>Simplify the denominator: $\sin^4 x + \cos^4 x = 1 - 2\sin^2 x \cos^2 x = 1 - \frac{\sin^2 2x}{2}$. The integral becomes $\int \frac{\sin 2x}{1 - \frac{\sin^2 2x}{2}} dx$. Using substitution and inverse trigonometric formulas, we get $a = -1, b = 1$.</p>
Correct Answer: B