Question:
The value of the integral \( \int_{0}^{\pi/2} \sin^6 x \cos^4 x \, dx \) is equal to:
The value of the integral \( \int_{0}^{\pi/2} \sin^6 x \cos^4 x \, dx \) is equal to:
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Wallis' Formula is the fastest "shortcut" for definite integrals of sine and cosine products with limits from \( 0 \) to \( \pi/2 \). Always check if both powers are even to decide the \( \pi/2 \) multiplier.
Updated On: May 20, 2026
- \( \frac{\pi}{256} \)
- \( \frac{\pi}{512} \)
- \( \frac{3\pi}{512} \)
- \( \frac{5\pi}{512} \)
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The Correct Option is C
Solution and Explanation
Concept:
To solve integrals of the form \( \int_{0}^{\pi/2} \sin^m x \cos^n x \, dx \), we useWallis' Formula (or the Gamma function approach):
• Formula: \( \frac{[(m-1)(m-3)\cdots 1 \text{ or } 2] \times [(n-1)(n-3)\cdots 1 \text{ or } 2]}{(m+n)(m+n-2)\cdots 1 \text{ or } 2} \times K \)
• If both \( m \) and \( n \) are even, \( K = \frac{\pi}{2} \).
• Otherwise, \( K = 1 \).
Step 1: Identifying parameters \( m \) and \( n \).
Given integral: \( \int_{0}^{\pi/2} \sin^6 x \cos^4 x \, dx \) Here, \( m = 6 \) and \( n = 4 \). Since both \( 6 \) and \( 4 \) areeven numbers, we will use \( K = \frac{\pi}{2} \).
Step 2: Applying the reduction formula values.
Numerator terms for \( m=6 \): \( (6-1)(6-3)(6-5) = 5 \cdot 3 \cdot 1 \) Numerator terms for \( n=4 \): \( (4-1)(4-3) = 3 \cdot 1 \) Denominator terms for \( m+n=10 \): \( 10 \cdot 8 \cdot 6 \cdot 4 \cdot 2 \)
Step 3: Calculating the final value.
\[ I = \frac{(5 \cdot 3 \cdot 1) \cdot (3 \cdot 1)}{10 \cdot 8 \cdot 6 \cdot 4 \cdot 2} \times \frac{\pi}{2} \] Simplifying the fractions: \[ I = \frac{15 \cdot 3}{3840} \times \frac{\pi}{2} = \frac{45}{3840} \times \frac{\pi}{2} \] Dividing 45 and 3840 by 15: \[ I = \frac{3}{256} \times \frac{\pi}{2} = \frac{3\pi}{512} \]
• Formula: \( \frac{[(m-1)(m-3)\cdots 1 \text{ or } 2] \times [(n-1)(n-3)\cdots 1 \text{ or } 2]}{(m+n)(m+n-2)\cdots 1 \text{ or } 2} \times K \)
• If both \( m \) and \( n \) are even, \( K = \frac{\pi}{2} \).
• Otherwise, \( K = 1 \).
Step 1: Identifying parameters \( m \) and \( n \).
Given integral: \( \int_{0}^{\pi/2} \sin^6 x \cos^4 x \, dx \) Here, \( m = 6 \) and \( n = 4 \). Since both \( 6 \) and \( 4 \) areeven numbers, we will use \( K = \frac{\pi}{2} \).
Step 2: Applying the reduction formula values.
Numerator terms for \( m=6 \): \( (6-1)(6-3)(6-5) = 5 \cdot 3 \cdot 1 \) Numerator terms for \( n=4 \): \( (4-1)(4-3) = 3 \cdot 1 \) Denominator terms for \( m+n=10 \): \( 10 \cdot 8 \cdot 6 \cdot 4 \cdot 2 \)
Step 3: Calculating the final value.
\[ I = \frac{(5 \cdot 3 \cdot 1) \cdot (3 \cdot 1)}{10 \cdot 8 \cdot 6 \cdot 4 \cdot 2} \times \frac{\pi}{2} \] Simplifying the fractions: \[ I = \frac{15 \cdot 3}{3840} \times \frac{\pi}{2} = \frac{45}{3840} \times \frac{\pi}{2} \] Dividing 45 and 3840 by 15: \[ I = \frac{3}{256} \times \frac{\pi}{2} = \frac{3\pi}{512} \]
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