Definite Integration
Substitution and Symmetry
Grade 12
Question:
<p><span class="math">\int_{0}^{1} f'(1-t) e^{-\cos \pi t} dt - \int_{1}^{2} f'(2-t) e^{\cos \pi t} dt</span> is equal to:</p>
<p>(a) <span class="math">\int_{0}^{2} f'(t) e^{\cos \pi t} dt</span></p>
<p>(b) 1</p>
<p>(c) 2</p>
<p>(d) <span class="math">\pi</span></p>
Step-by-Step Solution
Key Concept: Use substitution and the symmetry properties of <span class="math">f'</span> to transform and combine the two integrals.
<p><strong>Solution:</strong> In the first integral, substitute <span class="math">s = 1-t</span> to get <span class="math">\int_{0}^{1} f'(s) e^{-\cos \pi (1-s)} (-ds) = -\int_{1}^{0} f'(s) e^{-\cos \pi + \cos \pi s} ds</span>. In the second integral, substitute <span class="math">u = 2-t</span>. Use the antisymmetry <span class="math">f'(x) = -f'(2-x)</span> and the periodicity of cosine to simplify and combine the integrals.</p>
Correct Answer: d