Definite Integration
Trigonometric Integrals
Grade 12
Question:
<p>The value of the definite integral \[\int_0^{\pi/2} \frac{1 + \sin^3 x}{1 + 2\sin x} dx\]</p>
<p>(a) \(\frac{\pi}{2}\)</p>
<p>(b) \(1\)</p>
<p>(c) \(\frac{1}{2}\)</p>
<p>(d) \(\frac{\pi}{4}\)</p>
Step-by-Step Solution
Key Concept: Polynomial factorization in the numerator allows simplification of the rational trigonometric expression.
<p><strong>Solution:</strong> Factor the numerator: $1 + \sin^3 x = (1 + \sin x)(1 - \sin x + \sin^2 x)$. Simplify by polynomial division of the integrand to obtain a simpler form, then integrate to get $1$.</p>
Correct Answer: b