Properties of Definite Integrals
Chapter 7: CBSE 2025-26 Compilation
1. Basic Properties (Change of Variable & Limits)
| Property | Formula |
|---|---|
| $$ P_0 $$: Change of Variable (Dummy variable) |
$$ \int_a^b f(x) \, dx = \int_a^b f(t) \, dt $$ |
| $$ P_1 $$: Interchanging Limits | $$ \int_a^b f(x) \, dx = – \int_b^a f(x) \, dx $$ |
| $$ P_2 $$: Splitting the Interval (where $$ a < c < b $$) |
$$ \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx $$ |
2. Reflection Properties (King’s Property)
| Property | Formula |
|---|---|
| $$ P_3 $$: King’s Property (Sum of limits – x) |
$$ \int_a^b f(x) \, dx = \int_a^b f(a + b – x) \, dx $$ |
| $$ P_4 $$: Special Case (Derived from $$ P_3 $$) |
$$ \int_0^a f(x) \, dx = \int_0^a f(a – x) \, dx $$ |
3. Symmetry & Even/Odd Functions
| Interval | Condition & Result |
|---|---|
| $$ P_6 $$: Interval $$ [0, 2a] $$ $$ \int_0^{2a} f(x) \, dx $$ |
If $$ f(2a – x) = f(x) $$: $$ = 2 \int_0^a f(x) \, dx $$ |
| If $$ f(2a – x) = -f(x) $$: $$ = 0 $$ |
|
| $$ P_7 $$: Interval $$ [-a, a] $$ $$ \int_{-a}^{a} f(x) \, dx $$ |
If Even Function ($$ f(-x) = f(x) $$): $$ = 2 \int_0^a f(x) \, dx $$ |
| If Odd Function ($$ f(-x) = -f(x) $$): $$ = 0 $$ |