Definite Integrals — Part 2: Reflection, symmetry, even and odd functions: visual lesson, ISC Class 12 Mathematics

Lesson

Definite Integrals, Part 2

Some integrals that have no easy antiderivative can still be evaluated exactly by symmetry.

In this lesson you will

Part 2 at a glance.

Concept

Each is proved by one substitution.
Property	Statement	Typical use
P3	$\displaystyle\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx$	limits that add up to the number inside
P4	$\displaystyle\int_0^a f(x)\,dx = \int_0^a f(a - x)\,dx$	swapping $\sin$ and $\cos$ on $\left[0, \tfrac{\pi}{2}\right]$
P5	$\displaystyle\int_0^{2a} f = \int_0^a f(x)\,dx + \int_0^a f(2a - x)\,dx$	symmetry about $x = a$
P6	$\displaystyle\int_{-a}^{a} f = 2\int_0^a f$ (even), $0$ (odd)	balanced intervals

Diagram

Reflecting in $x = \tfrac{a + b}{2}$ does not change the integral.

$\displaystyle\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx$

\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx