Definite Integration
Integration by Substitution
Grade 12

Question:

<p>∫₀ᵖ e^(sin 2x) cos 3x dx is equal to</p>
<p>(a) -1</p>
<p>(b) 0</p>
<p>(c) 1</p>
<p>(d) π</p>

Step-by-Step Solution

Key Concept: Use substitution x → π - x to reveal that the integrand has an odd symmetry property about the limits.
<p>Let I = ∫₀ᵖ e^(sin 2x) cos 3x dx. Substitute x → π - x: I = -∫₀ᵖ e^(sin 2(π-x)) cos 3(π-x) dx = -∫₀ᵖ e^(sin 2x) × (-cos 3x) dx = ∫₀ᵖ e^(sin 2x) cos 3x dx = I. This gives 2I = 0, so I = 0.</p>
Correct Answer: B

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