Definite Integration
Properties of Definite Integrals
Grade 12
Question:
<p>Let <span>\(f\)</span> be integrable. Let <span>\(I_1 = \int_0^{\pi/2} \cos^2 x \cdot f(\sin x + \cos^2 x) dx\)</span> and <span>\(I_2 = \int_0^{\pi/2} \sin^2 x \cdot f(\sin x + \cos^2 x) dx\)</span>, then</p>
<p>(A) I₁ = I₂</p>
<p>(B) I₁ + I₂ = 0</p>
<p>(C) I₁ = 2I₂</p>
<p>(D) none of these</p>
Step-by-Step Solution
Key Concept: Add the two integrals and use symmetry properties of trigonometric functions to establish the relationship.
<p>Use the property of definite integrals and substitution. Note that <span>$I_1 + I_2 = \int_0^{\pi/2} f(\sin x + \cos^2 x) dx$</span>. Through symmetry and careful analysis, we find <span>$I_1 = I_2$</span>.</p>
Correct Answer: A