<p>Let <span class="math">\(I_1 = \int_0^1 x \times x^{49}(1-x^{50})^{100} dx\)</span>. If <span class="math">\(I_2 = aI_1\)</span>, find the value of <span class="math">\(a\)</span>.</p>
Step-by-Step Solution
Key Concept: Use substitution and integration by parts to relate two integrals, then solve for the coefficient relating them.
<p><strong>Step 1:</strong> Use integration by parts with <span class="math">$u = x$</span> and <span class="math">$dv = x^{49}(1-x^{50})^{100} dx$</span>.</p><p><strong>Step 2:</strong> Substitute <span class="math">$x^{50} = t$</span>, so <span class="math">$50x^{49}dx = dt$</span>.</p><p><strong>Step 3:</strong> The integral becomes <span class="math">$I_1 = \frac{1}{50 \times 101}$</span> after evaluation.</p><p><strong>Step 4:</strong> Calculate <span class="math">$I_2 = I_1 - \frac{I_2}{50 \times 101}$</span>.</p><p><strong>Step 5:</strong> Solving for the relationship: <span class="math">$I_2 = \frac{5050}{5051}I_1$</span>.</p><p>∴ <span class="math">$a = \frac{5050}{5051}$</span></p>
Correct Answer: 5050/5051