Step 1: Understanding the Concept:
Evaluating integrals of the form \(\int_{0}^{\pi/2} \sin^m x \cos^n x dx\) is a standard task in engineering mathematics.
Direct integration using trigonometric expansion (like power-reduction formulas) is extremely tedious and prone to error.
Instead, we use Wallis’ Formula, which is a reduction-formula-based shortcut specifically designed for limits from \(0\) to \(\pi/2\).
This formula converts the integral into a ratio of products of integers.
The formula differs slightly depending on whether the exponents \(m\) and \(n\) are even or odd.
Key Formula or Approach:
Wallis’ Formula:
\[ \int_{0}^{\pi/2} \sin^m x \cos^n x dx = \frac{[(m-1)(m-3)\dots][(n-1)(n-3)\dots]}{(m+n)(m+n-2)\dots} \cdot K \]
Where:
- \(K = \frac{\pi}{2}\) if both \(m\) and \(n\) are even.
- \(K = 1\) if at least one of \(m\) or \(n\) is odd.
The product in the numerator and denominator stops when it reaches 1 or 2.
Step 2: Detailed Explanation:
Identify the parameters: \(m = 6\) and \(n = 4\).
Since both 6 and 4 are even numbers, our multiplier \(K\) will be \(\frac{\pi}{2}\).
Calculate the numerator sequence for \(m=6\):
\((m-1)(m-3)(m-5) = 5 \cdot 3 \cdot 1 = 15\).
Calculate the numerator sequence for \(n=4\):
\((n-1)(n-3) = 3 \cdot 1 = 3\).
Calculate the denominator sequence for \(m+n=10\):
\((m+n)(m+n-2)(m+n-4)(m+n-6)(m+n-8) = 10 \cdot 8 \cdot 6 \cdot 4 \cdot 2\).
\[ 10 \cdot 8 \cdot 6 \cdot 4 \cdot 2 = 3840 \]
Now, substitute everything into the formula:
\[ I = \frac{15 \cdot 3}{3840} \cdot \frac{\pi}{2} \]
\[ I = \frac{45}{3840} \cdot \frac{\pi}{2} \]
To simplify \(\frac{45}{3840}\), we can divide both numerator and denominator by 15:
\(45 \div 15 = 3\).
\(3840 \div 15 = 256\).
So the simplified fraction is \(\frac{3}{256}\).
Finally:
\[ I = \frac{3}{256} \cdot \frac{\pi}{2} = \frac{3\pi}{512} \]
Step 3: Final Answer:
The result of the integral using Wallis' Formula is \(\frac{3\pi}{512}\).
This matches option (C).