Definite Integration of Odd and Even Functions

Definite Integration of Odd and Even Functions

How to solve definite integration of odd and even functions problems: definitions, formulas, examples, and their solutions.

Odd Functions

The graph of a odd function is symmetric with respect to the origin. Examples: x, x^3, x^5, x^7, sin x, tan x

An odd function is a function
whose graph is symmetric with respect to the origin.

Here are some examples:
x(odd): x, x3, x5, x7, ...
sin x, tan x
sinh x, tanh x

Graphing polynomial functions - Odd power functions

Graphing sine functions

Graphing tangent functions

Even Functions

The graph of an even function is symmetric with respect to the y-axis. Examples: C, x^2, x^4, x^6, cos x

An even function is a function
whose graph is symmetric with respect to the y-axis.

Here are some examples:
x(even): C (= Cx0), x2, x4, x6, ...
cos x
cosh x

Graphing polynomial functions - Even power functions

Graphing cosine functions

Definite Integration of Odd Functions

[inteval, from -a to a, of f(x) dx] = 0

-aa (odd) dx = 0

The left area is under the x-axis.

So, if the integral of the right half is S,
then the integral of the left half is -S.

So the whole integral is
-S + S = 0.

Definite Integration of Even Functions

[inteval, from -a to a, of f(x) dx] = 2*[inteval, from 0 to a, of f(x) dx]

-aa (even) dx = 0

If the integral of the right half is S,
then the integral of the left half is the same: S.

So the whole integral is
S + S = 2S
= ∫0a (even) dx.

Example 1

Find the given integral. The integral, from -1 to 1, of (5x^4 + 8x^3 - 6x^2 - 2x + 7) dx

The interval of the integral is symmetric: [-1, 1].

So multiply 2,
change the interval to [0, 1],
and cancel the odd functions (dark gray).

Solve the definite integral.

Definite integration of polynomials

Example 2

Find the given integral. The integral, from -pi/3 to pi/3, of (sin theta + cos theta) d theta

The interval of the integral is symmetric: [-π/3, π/3].

So multiply 2,
change the interval to [0, π/3],
and cancel the odd function sin θ.

Solve the definite integral.

Indefinite integration of cos x

Example 3

Find the given integral. The integral, from -pi/6 to pi/6, of (sin theta + cos theta)^2 d theta

Square of a sum

sin2 θ + cos2 θ = 1

Pythagorean identities

2 sin θ cos θ = 2 sin 2θ

sin 2A (Double-angle formula)

The interval of the integral is symmetric: [-π/6, π/6].

So multiply 2,
change the interval to [0, π/6],
and cancel the odd function sin 2θ.

Solve the definite integral.

Example 4

Find the given integral. The integral, from -pi/4 to pi/4, of (tan theta) d theta

The interval of the integral is symmetric: [-π/4, π/4].
And tan θ is an odd function.

So this interval is 0.