<p>Evaluate \(\displaystyle\int_0^{\pi/2} \sin^5 x\cos^4 x\,dx\)</p>
Step-by-Step Solution
Key Concept: Wallis formula: \int_0^(\pi/2) sinᵐx \cdot cosⁿx dx = [(m-1)\!\! \cdot (n-1)\!\!] / (m+n)\!\! \cdot factor
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<p><strong>Using Wallis formula:</strong> \(\displaystyle\int_0^{\pi/2}\sin^m x\cos^n x\,dx = \frac{(m-1)\!\!(n-1)\!\!}{(m+n)\!\!}\) when \(m+n\) is odd, no extra factor.</p>
<p>Here \(m=5\), \(n=4\): \(m+n=9\) (odd) \to no \(\pi/2\) factor.</p>
<p>\((m-1)\!\! = 4\!\! = 4\cdot2 = 8\)</p>
<p>\((n-1)\!\! = 3\!\! = 3\cdot1 = 3\)</p>
<p>\((m+n)\!\! = 9\!\! = 9\cdot7\cdot5\cdot3\cdot1 = 945\)</p>
<p>\[I = \frac{8\cdot3}{945} = \frac{24}{945} = \frac{8}{315}\]</p>
<p><strong>Verification by substitution:</strong> Let \(t=\cos x\):</p>
<p>\[I = \int_0^1(1-t^2)^2 t^4\,dt = \int_0^1(t^4-2t^6+t^8)\,dt = \frac{1}{5}-\frac{2}{7}+\frac{1}{9} = \frac{63-90+35}{315} = \frac{8}{315}\checkmark\]</p>
Correct Answer: B