<p><span class="math">\int_{-1}^{1} f'(1+x^2) x^2 e^{-\cos \pi t} dx</span> is equal to:</p>
Step-by-Step Solution
Key Concept: Recognize the antisymmetry property of <span class="math">f'</span> derived from <span class="math">f(x) = f(2-x)</span> and verify that the integrand is odd.
<p><strong>Solution:</strong> Let <span class="math">u = 1 + x^2</span>, so the integrand involves <span class="math">f'(u)</span>. From the given conditions on <span class="math">f</span> (that <span class="math">f(x) = f(2-x)</span>), we have <span class="math">f'(x) = -f'(2-x)</span>. The substitution <span class="math">x \to -x</span> shows that the integrand is an odd function, hence the integral equals 0.</p>
Correct Answer: d