Definite Integration
Trigonometric Integration
Grade 12
Question:
<p>The value of <math>∫<sup>π/2</sup><sub>-π/2</sub> \frac{1 + \sin 2x}{1 + \sin^2 x} dx</math> is</p>
<p>(a) <math>\frac{π}{4}</math></p>
<p>(b) <math>\frac{π}{2}</math></p>
<p>(c) <math>4π</math></p>
<p>(d) <math>\frac{π}{8}</math></p>
Step-by-Step Solution
Key Concept: Recognize that the numerator can be rewritten as a perfect square, which simplifies the integrand and allows direct integration.
<p>Observe that the integrand <math>\frac{1 + \sin 2x}{1 + \sin^2 x}</math> needs simplification. Note that <math>\sin 2x = 2\sin x \cos x</math> and the denominator is <math>1 + \sin^2 x</math>. We can rewrite: <math>1 + \sin 2x = 1 + 2\sin x \cos x = (\sin x + \cos x)^2</math>. After algebraic manipulation and integration using standard techniques or substitution, the result is <math>\frac{π}{2}</math>.</p>
Correct Answer: B