<p>Evaluate \(\displaystyle\int_0^1 x^4(1-x)^3\,dx\)</p>
Step-by-Step Solution
Key Concept: Beta function: B(m+1,n+1) = m\! \cdot n\!/(m+n+1)\! for non-negative integers m,n.
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<p>\(\displaystyle\int_0^1 x^m(1-x)^n\,dx = B(m+1,n+1) = \frac{m\!\,n\!}{(m+n+1)\!}\)</p>
<p>Here \(m=4,\,n=3\):</p>
<p>\[I = \frac{4\!\cdot 3\!}{8\!} = \frac{24\cdot 6}{40320} = \frac{144}{40320} = \frac{1}{280}\]</p>
<p><strong>Verification:</strong> Expand \((1-x)^3 = 1-3x+3x^2-x^3\):</p>
<p>\[\int_0^1(x^4-3x^5+3x^6-x^7)\,dx = \frac{1}{5}-\frac{3}{6}+\frac{3}{7}-\frac{1}{8} = \frac{1}{5}-\frac{1}{2}+\frac{3}{7}-\frac{1}{8}\]</p>
<p>\[= \frac{56-140+120-35}{280} = \frac{1}{280}\checkmark\]</p>
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Correct Answer: C