Definite Integration
Properties of Definite Integrals
Grade 12
Question:
<p>Let <i>f</i> be integrable over [0, <i>a</i>] for any real <i>a</i>. If we define<br/><i>I</i><sub>1</sub> = ∫₀^(π/2) cos θ <i>f</i>(sin θ + cos θ) <i>d</i>θ<br/>and <i>I</i><sub>2</sub> = ∫₀^(π/2) sin² θ <i>f</i>(sin θ + cos² θ) <i>d</i>θ<br/>then the relationship between <i>I</i><sub>1</sub> and <i>I</i><sub>2</sub> is:</p>
<p>(a) <i>I</i><sub>1</sub> = <i>I</i><sub>2</sub></p>
<p>(b) <i>I</i><sub>1</sub> = −<i>I</i><sub>2</sub></p>
<p>(c) <i>I</i><sub>1</sub> = 2<i>I</i><sub>2</sub></p>
<p>(d) <i>I</i><sub>1</sub> = −2<i>I</i><sub>2</sub></p>
Step-by-Step Solution
Key Concept: Use substitution and properties of definite integrals to establish relationship between integrals with transformed functions.
<p>Apply the property of definite integrals and use substitution techniques to relate the two integrals.</p>
Correct Answer: A