The Wallis Formula: Integrals of Powers of Sine & Cosine

Why You Need the Wallis Formula

Traditional methods for evaluating $\int \sin^n x dx$ and $\int \cos^n x dx$ require:

Repeated integration by parts - Time consuming and error-prone
Complex trigonometric identities - Hard to remember and apply
Lengthy calculations - Wastes precious exam time
Higher chance of mistakes - Each step introduces potential errors

The Wallis Formula gives you a direct result without these complications.

🎯 JEE Exam Advantage

Wallis Formula appears in definite integrals, area problems, and volume calculations. Mastering it can save you 5-10 minutes per problem in JEE Main and Advanced.

1. Understanding the Wallis Formula

The Wallis Formula (Definite Integrals)

For positive integers n:

$$ \int_0^{\frac{\pi}{2}} \sin^n x dx = \int_0^{\frac{\pi}{2}} \cos^n x dx = \begin{cases} \frac{(n-1)!!}{n!!} \cdot \frac{\pi}{2} & \text{if } n \text{ is even} \\ \frac{(n-1)!!}{n!!} & \text{if } n \text{ is odd} \end{cases} $$

Where $n!!$ denotes the double factorial:

$n!! = n \cdot (n-2) \cdot (n-4) \cdots $ (until 2 or 1)
Example: $6!! = 6 \cdot 4 \cdot 2 = 48$
Example: $7!! = 7 \cdot 5 \cdot 3 \cdot 1 = 105$

Simplified Form (Easy to Remember)

Even Powers:

$$ \int_0^{\frac{\pi}{2}} \sin^{2m} x dx = \frac{(2m-1)!!}{(2m)!!} \cdot \frac{\pi}{2} $$

Odd Powers:

$$ \int_0^{\frac{\pi}{2}} \sin^{2m+1} x dx = \frac{(2m)!!}{(2m+1)!!} $$

2. Step-by-Step Examples

Example 1: Even Power (n = 4)

Evaluate $\int_0^{\frac{\pi}{2}} \sin^4 x dx$

Step 1: Identify n = 4 (even)

Step 2: Apply formula for even powers:

$$ \int_0^{\frac{\pi}{2}} \sin^4 x dx = \frac{(4-1)!!}{4!!} \cdot \frac{\pi}{2} = \frac{3!!}{4!!} \cdot \frac{\pi}{2} $$

Step 3: Calculate double factorials:

$3!! = 3 \cdot 1 = 3$

$4!! = 4 \cdot 2 = 8$

Step 4: Substitute:

$$ = \frac{3}{8} \cdot \frac{\pi}{2} = \frac{3\pi}{16} $$

✅ Final Answer: $\frac{3\pi}{16}$

Example 2: Odd Power (n = 5)

Evaluate $\int_0^{\frac{\pi}{2}} \cos^5 x dx$

Step 1: Identify n = 5 (odd)

Step 2: Apply formula for odd powers:

$$ \int_0^{\frac{\pi}{2}} \cos^5 x dx = \frac{(5-1)!!}{5!!} = \frac{4!!}{5!!} $$

Step 3: Calculate double factorials:

$4!! = 4 \cdot 2 = 8$

$5!! = 5 \cdot 3 \cdot 1 = 15$

Step 4: Substitute:

$$ = \frac{8}{15} $$

✅ Final Answer: $\frac{8}{15}$

3. Quick Reference Table

Integral	Value	Time Saved
$\int_0^{\frac{\pi}{2}} \sin^2 x dx$	$\frac{\pi}{4}$	2-3 min
$\int_0^{\frac{\pi}{2}} \sin^3 x dx$	$\frac{2}{3}$	3-4 min
$\int_0^{\frac{\pi}{2}} \sin^4 x dx$	$\frac{3\pi}{16}$	4-5 min
$\int_0^{\frac{\pi}{2}} \sin^5 x dx$	$\frac{8}{15}$	5-6 min
$\int_0^{\frac{\pi}{2}} \sin^6 x dx$	$\frac{5\pi}{32}$	6-7 min

💡 Memory Aid

Even powers: Result has $\pi$ and pattern: $\frac{1}{2}\cdot\frac{\pi}{2}, \frac{1\cdot3}{2\cdot4}\cdot\frac{\pi}{2}, \frac{1\cdot3\cdot5}{2\cdot4\cdot6}\cdot\frac{\pi}{2}, \dots$

Odd powers: No $\pi$, pattern: $\frac{2}{3}, \frac{2\cdot4}{3\cdot5}, \frac{2\cdot4\cdot6}{3\cdot5\cdot7}, \dots$

4. Special Cases & Extensions

Mixed Powers: $\int_0^{\frac{\pi}{2}} \sin^m x \cos^n x dx$

The Wallis Formula extends to mixed powers:

$$ \int_0^{\frac{\pi}{2}} \sin^m x \cos^n x dx = \frac{\Gamma\left(\frac{m+1}{2}\right) \Gamma\left(\frac{n+1}{2}\right)}{2 \Gamma\left(\frac{m+n+2}{2}\right)} $$

When m and n are positive integers:

$$ = \frac{(m-1)!! (n-1)!!}{(m+n)!!} \times \begin{cases} 1 & \text{if both m,n odd} \\ \frac{\pi}{2} & \text{otherwise} \end{cases} $$

Different Limits

Symmetric limits:

$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^n x dx = \begin{cases} 0 & \text{n odd} \\ 2\int_0^{\frac{\pi}{2}} \sin^n x dx & \text{n even} \end{cases}$

Full period:

$\int_0^{2\pi} \sin^n x dx = \begin{cases} 0 & \text{n odd} \\ 4\int_0^{\frac{\pi}{2}} \sin^n x dx & \text{n even} \end{cases}$

5. Common Mistakes to Avoid

⚠️ Critical Errors Students Make

Mistake 1: Forgetting the $\frac{\pi}{2}$ factor

For even powers, students often calculate $\frac{(n-1)!!}{n!!}$ but forget to multiply by $\frac{\pi}{2}$

Mistake 2: Confusing even and odd cases

Applying the even power formula to odd powers (or vice versa) gives completely wrong results

Mistake 3: Incorrect double factorial calculation

$n!!$ is NOT $(n!)!$. Remember: $n!! = n \cdot (n-2) \cdot (n-4) \cdots$

Mistake 4: Using wrong limits

Wallis Formula specifically applies to $[0, \frac{\pi}{2}]$. For other limits, you need to adjust accordingly

6. Practice Problems

Test Your Understanding

1. $\int_0^{\frac{\pi}{2}} \sin^6 x dx$

Hint: n=6 (even), pattern: $\frac{1\cdot3\cdot5}{2\cdot4\cdot6}\cdot\frac{\pi}{2}$

2. $\int_0^{\frac{\pi}{2}} \cos^7 x dx$

Hint: n=7 (odd), pattern: $\frac{2\cdot4\cdot6}{3\cdot5\cdot7}$

3. $\int_0^{\frac{\pi}{2}} \sin^4 x \cos^2 x dx$

Hint: Use the mixed power formula with m=4, n=2

4. $\int_0^{\pi} \sin^5 x dx$

Hint: Use symmetry: $\int_0^{\pi} \sin^n x dx = 2\int_0^{\frac{\pi}{2}} \sin^n x dx$ for odd n

Answers:

1. $\frac{5\pi}{32}$

2. $\frac{16}{35}$

3. $\frac{\pi}{32}$

4. $\frac{16}{15}$

🎯 JEE Exam Strategy

When to Use Wallis Formula:

Definite integrals with limits $0$ to $\frac{\pi}{2}$
Powers of sine or cosine (or both)
Area under trigonometric curves
Volume of revolution problems
Fourier series coefficients

Time-Saving Approach:

Memorize patterns for n=2 to n=6
Practice mental calculation of double factorials
Recognize symmetric limits quickly
Combine with substitution for complex integrals

7. Quick Revision Sheet

Key Formulas

$\int_0^{\frac{\pi}{2}} \sin^{2m} x dx = \frac{(2m-1)!!}{(2m)!!} \cdot \frac{\pi}{2}$

$\int_0^{\frac{\pi}{2}} \sin^{2m+1} x dx = \frac{(2m)!!}{(2m+1)!!}$

Same for $\cos^n x$

$\int_0^{\frac{\pi}{2}} \sin^m x \cos^n x dx = \frac{(m-1)!!(n-1)!!}{(m+n)!!} \times K$

Common Values

$\sin^2, \cos^2$: $\frac{\pi}{4}$

$\sin^3, \cos^3$: $\frac{2}{3}$

$\sin^4, \cos^4$: $\frac{3\pi}{16}$

$\sin^5, \cos^5$: $\frac{8}{15}$

$\sin^6, \cos^6$: $\frac{5\pi}{32}$

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