Walli's Formula | JEE Main Calculus Crash Course | Mathematicon

Walli's Formula: A Shortcut for Definite Integrals

Hey JEE aspirants! Definite integrals can be tricky, but Walli's formula offers a neat shortcut for evaluating integrals of the form $\int_0^{\pi/2} \sin^n x \, dx$ or $\int_0^{\pi/2} \cos^n x \, dx$ . Mastering this formula can save you precious time in the exam, especially for objective-type questions. Let's dive in!

Understanding the Concept

Walli's formula provides a direct way to compute the definite integral of $\sin^n x$ or $\cos^n x$ from $0$ to $\pi/2$ . The beauty lies in recognizing the pattern based on whether $n$ is even or odd. Instead of grinding through integration by parts repeatedly, you can plug the value of $n$ into a simple expression.

Intuition: Imagine the area under the curve of $\sin^n x$ (or $\cos^n x$ ) between $0$ and $\pi/2$ . As $n$ increases, the curve gets "squished" closer to the x-axis, especially near $0$ and $\pi/2$ , making the area smaller. Walli's formula captures this shrinking behavior using a product of fractions.

The Formulas

First, remember this fundamental property:

\int_0^{\pi/2} \sin^n x \, dx = \int_0^{\pi/2} \cos^n x \, dx

This is because the sine and cosine curves are just phase-shifted versions of each other. Now, let's break down Walli's formula for even and odd values of $n$ .

Case 1: n is Even

\int_0^{\pi/2} \sin^n x \, dx = \frac{(n-1)(n-3) \cdots 3 \cdot 1}{n(n-2) \cdots 4 \cdot 2} \times \frac{\pi}{2}

Explanation: Notice that we multiply all odd numbers from $1$ to $n-1$ in the numerator and all even numbers from $2$ to $n$ in the denominator, then multiply by $\frac{\pi}{2}$ . For example:

\int_0^{\pi/2} \sin^4 x \, dx = \frac{3 \cdot 1}{4 \cdot 2} \times \frac{\pi}{2} = \frac{3\pi}{16}

Case 2: n is Odd

\int_0^{\pi/2} \sin^n x \, dx = \frac{(n-1)(n-3) \cdots 4 \cdot 2}{n(n-2) \cdots 5 \cdot 3 \cdot 1}

Explanation: The pattern is similar to the even case, but without the $\frac{\pi}{2}$ factor. Again, we multiply odd numbers in the denominator and even numbers in the numerator, but the ranges are slightly different. For example:

Let's evaluate $\int_0^{\pi/2} \sin^5 x \, dx$ . Here, $n = 5$ (odd). Applying Walli's formula:

$\int_0^{\pi/2} \sin^5 x \, dx = \frac{4 \cdot 2}{5 \cdot 3 \cdot 1} = \frac{8}{15}$

Tips for Solving Problems

Tip 1: Always check if the limits of integration are from $0$ to $\pi/2$ . Walli's formula is only applicable for these limits.

Tip 2: Memorize the formula for both even and odd cases. This will save you valuable time during the exam.

Tip 3: Practice, practice, practice! The more you apply the formula, the more comfortable you'll become with it.

Common Mistakes to Avoid

Mistake 1: Forgetting the $\frac{\pi}{2}$ factor when $n$ is even.

Mistake 2: Confusing the numerator and denominator when $n$ is odd.

Mistake 3: Applying Walli's formula when the limits of integration are not $0$ and $\pi/2>.

JEE-Specific Tricks

Trick 1: Walli's formula often appears indirectly in JEE problems. Look for integrals that can be transformed into the form $\int_0^{\pi/2} \sin^n x \, dx$ or $\int_0^{\pi/2} \cos^n x \, dx$ using suitable substitutions.

Trick 2: Sometimes, JEE problems may involve a series of integrals with varying powers of sine or cosine. Use Walli's formula to quickly evaluate each integral and then sum the results.

Applications

Walli's formula is not just a theoretical concept; it's a powerful tool with applications in various fields:

Physics: Calculating the average value of powers of trigonometric functions in wave mechanics.
Engineering: Determining the root mean square (RMS) value of sinusoidal signals.
Mathematics: Evaluating complex definite integrals and infinite products.

By understanding Walli's formula and practicing its application, you'll be well-equipped to tackle a wide range of JEE problems involving definite integrals. Keep practicing, and you'll ace those integrals!