Description:
In this post, we discuss some important properties of definite integrals which simplify calculations and make evaluation easier. These properties are frequently used in NCERT Class 12 Mathematics and are very useful for solving integration problems in board exams and competitive exams like JEE.
Introduction
The definite integral not only represents the area under a curve but also satisfies several useful properties. These properties allow us to transform, split, or simplify integrals, making problem-solving much faster. Below, we list the most important properties of definite integrals along with their standard forms.
Properties
Property P₀: \int_a^b f(x),dx = \int_a^b f(t),dt
(This shows that the variable of integration is dummy and can be replaced.)
Property P₁: \int_a^b f(x) \ dx = -\int_b^a f(x) \ dx
particular case: \int_a^a f(x) \ dx = 0
Property P₂: \int_a^b f(x) \ dx = \int_a^c f(x) \dx + \int_c^b f(x) \ dx
(This allows splitting an integral at an intermediate point.)
Property P₃: \int_a^b f(x) \ dx = \int_a^b f(a+b-x) \ dx
(This is useful for symmetry-based simplification.)
Property P₄: \int_0^a f(x) \ dx = \int_0^a f(a-x) \ dx
(Note: This is a special case of Property P₃.)
Property P₅: \int_0^{2a} f(x) \ dx = \int_0^a f(x) \ dx + \int_0^a f(2a-x) \ dx
Property P₆:
\int_{0}^{2a} f(x) \ dx = 2 \int_{0}^{a} f(x) \ dx, \ \ \ \ \text{if } f(2a-x) = f(x) and
\int_{0}^{2a} f(x) \ dx = 0, \ \ \ \ \text{if } f(2a-x) = -f(x)
Property P₇:
\int_{-a}^{a} f(x),dx = 2 \int_{0}^{a} f(x),dx, \ \ \ \ \text{if } f \text{ is an even function, i.e., } f(-x) = f(x) and
\int_{-a}^{a} f(x),dx = 0, \ \ \ \ \text{if } f \text{ is an odd function, i.e., } f(-x) = -f(x)
Conclusion
These properties form the foundation of many definite integral problems. By applying them strategically, one can simplify complex integrals without evaluating them directly. They are especially useful for problems involving symmetry or where direct integration is difficult.
For formulas of indefinite integration go through the following link
Formulas for indefinite integration
For complete list of class 12 Maths important formulas go through the following link
Class 12 Maths Important Formula