# Nakafa Framework: LLM

URL: /en/subject/high-school/12/mathematics/integral/properties-of-definite-integral
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/12/mathematics/integral/properties-of-definite-integral/en.mdx

Output docs content for large language models.

---

export const metadata = {
    title: "Properties of Definite Integral",
    description: "Master essential properties of definite integrals: identical limits, reversing bounds, constant factors, sum rules, and interval splitting techniques.",
    authors: [{ name: "Nabil Akbarazzima Fatih" }],
    date: "05/26/2025",
    subject: "Integrals",
};

## Identical Integration Limits

If the upper and lower limits of a definite integral are the same, the result is zero.

<BlockMath math="\int_{a}^{a} f(x) \, dx = 0" />

This makes a lot of sense intuitively. Since a definite integral calculates the area under a curve over an interval, if the interval has no width (from <InlineMath math="a" /> to <InlineMath math="a" />), then there is no area to calculate. It's like trying to find the area of a straight line, which is of course zero.

## Reversing the Integration Limits

When we swap the lower and upper limits of an integral, the result is the negative of the original integral's value.

<BlockMath math="\int_{a}^{b} f(x) \, dx = - \int_{b}^{a} f(x) \, dx" />

A simple analogy is measuring displacement. The distance from point A to B is the same as the distance from B to A, but the direction is opposite. The negative sign here represents the opposite "direction" in the context of integration.

## Constant Multiple Rule

Just as with indefinite integrals, a constant can be factored out of the integral to simplify the calculation.

<BlockMath math="\int_{a}^{b} kf(x) \, dx = k \int_{a}^{b} f(x) \, dx" />

If we have a function that is "scaled up" or "scaled down" by a constant factor <InlineMath math="k" />, we can calculate its base area first (<InlineMath math="\int f(x)dx" />) and then multiply the result by that factor <InlineMath math="k" />.

## Sum and Difference Rule

The integral of a sum or difference of two functions is equal to the sum or difference of their individual integrals.

<div className="flex flex-col gap-4">
<BlockMath math="\int_{a}^{b} [f(x) + g(x)] \, dx = \int_{a}^{b} f(x) \, dx + \int_{a}^{b} g(x) \, dx" />
<BlockMath math="\int_{a}^{b} [f(x) - g(x)] \, dx = \int_{a}^{b} f(x) \, dx - \int_{a}^{b} g(x) \, dx" />
</div>

This property allows us to break down the integral of a complex function into several simpler integrals. We can calculate the area under <InlineMath math="f(x)" /> and <InlineMath math="g(x)" /> separately, and then add or subtract them.

## Interval Addition Property

An integration interval can be split into several sub-intervals. The total area over the large interval is equal to the sum of the areas of its constituent sub-intervals.

<BlockMath math="\int_{a}^{c} f(x) \, dx = \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx" />

This property holds regardless of the order of <InlineMath math="a" />, <InlineMath math="b" />, and <InlineMath math="c" />. It's like saying that the journey from city A to city C is the same as the journey from A to B plus the journey from B to C. This is a very useful tool for handling functions that are defined differently on different intervals.