5 Definite Integral Properties Every JEE/Calculus Student Must Know

Mastering calculus can feel like trying to solve a puzzle where the pieces keep changing shape. If you’re preparing for the JEE, AP Calculus, or even advanced SAT Math, you’ve likely encountered definite integrals. While many students think definite integration is just “indefinite integration plus plugging in numbers,” it is actually much more powerful than that.

In fact, some integrals are impossible to solve using standard antiderivatives but become effortless when you apply the right property. In this blog, we will see  5 essential definite integral properties that will turn you into a calculus pro.

What are definite integrals?

The definite integral is defined as an integral with two specified limits called the upper and the lower limit. The definite integral of a function generally represents the area under the curve from the lower bound value to the higher bound value.

(I) King’s Property (The Game Changer)

This is arguably the most important property for JEE. It is often called the “King’s Rule” because of how frequently it solves “impossible” problems. It states that the integral remains unchanged if you replace x  with the sum of the limits minus x. 

The rule: \(\displaystyle \int_{a}^{b} f(x)\,dx = \int_{a}^{b} f(a + b – x)\,dx\)

It allows you to replace every x in the function with the sum of the limits minus x. This is particularly effective for removing “unwanted x” terms that are stuck in front of trigonometric functions. By adding the original integral and the “King’s” version together, the difficult parts often cancel out, leaving you with a simple integration.

Pro Tip: Use this when you see trigonometric functions like sin x and cos x in the same integrand to simplify them into a constant.

Recommended Reading: How to Use King’s Rule in Definite Integrals: Formulas & Solved Examples

(II) Even and Odd Functions (The Symmetry Rule)

Before you start a long calculation, always look at the limits. If you see limits going from a negative number to its positive counterpart (like −5 to 5), stop and check if the function is even or Odd.

Odd Functions: If f(−x)=−f(x) (like sin(x) or x³), the integral is zero. The areas on the left and right sides of the y-axis perfectly cancel each other out.

\(\displaystyle \int_{-a}^{a} f(x)\,dx = 0\)

Even Functions: If f(−x)=f(x) (like cos(x) or x²), the area is symmetrical. You can just calculate the integral from 0 to a and double it.

\(\displaystyle \int_{-a}^{a} f(x)\,dx = 2 \int_{0}^{a} f(x)\,dx\)

Pro-Tip for JEE/SAT: Spotting an odd function can save you five minutes of unnecessary work—you can just write “0” and move on.

Recommended Reading: The Wallis Formula: Integrating Powers of Sine and Cosine Instantly

(III) The “Break-Point” Property: Handling Absolute Values

Sometimes, a function behaves differently in different sections. For example, a modulus (absolute value) function changes its “rule” at zero. To solve these, You can split an integral into parts using point c within the interval [a,b] 

The Rule: \(\displaystyle \int_{a}^{b} f(x)\,dx = \int_{a}^{c} f(x)\,dx + \int_{c}^{b} f(x)\,dx\)

Pro Tip: This is a lifesaver for SAT and JEE students dealing with functions like ∣x−2∣ or the Greatest Integer Function. You simply find the “break point” where the function changes (like x=2) and split the integral there to solve each part separately.

Recommended Reading: Mastering Integrals with Modulus and Greatest Integer Functions

(IV) Reversal of Limits (The Sign Flip)

Swapping the upper and lower limits changes the sign of the integral. What happens if you need to swap the top and bottom numbers (the limits) of your integral? In definite integration, the order matters. If you interchange the lower limit (a) and the upper limit (b), you must multiply the entire integral by −1

The Rule: \(\displaystyle \int_{a}^{b} f(x)\,dx = – \int_{b}^{a} f(x)\,dx\)

Pro tip: If the upper and lower limits are the same (e.g., integrating from 1 to 1), the area under the curve is zero because there is no “width” to the interval.

(V) Queen’s Property (The Limit Reducer)

The “Queen’s Property” in definite integrals, often synonymous with the King’s Property, is a reflection formula used to simplify complex integrations. It’s a property designed to handle integrals where the upper limit (let’s call it 2a) can be split in half.

The Rule: \(\displaystyle
\int_{0}^{2a} f(x)\,dx
= \int_{0}^{a} \bigl[f(x) + f(2a – x)\bigr]\,dx
\)

• If f(2a – x) = f(x), then the integral becomes \(\displaystyle 2 \int_{0}^{a} f(x)\,dx\)
• If f(2a – x) = -f(x), then the integral is zero.

Recommended Reading: How to Use the Queen’s Property in Integrals