# 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

An odd function is a function

whose graph is symmetric with respect to the origin.

Here are some examples:*x*^{(odd)}: *x*, *x*^{3}, *x*^{5}, *x*^{7}, ...

sin *x*, tan *x*

sinh *x*, tanh *x*

Graphing polynomial functions - Odd power functions

Graphing sine functions

Graphing tangent functions

## Even Functions

An even function is a function

whose graph is symmetric with respect to the *y*-axis.

Here are some examples:*x*^{(even)}: *C* (= *C**x*^{0}), *x*^{2}, *x*^{4}, *x*^{6}, ...

cos *x*

cosh *x*

Graphing polynomial functions - Even power functions

Graphing cosine functions

## Definite Integration of Odd Functions

∫_{-a}^{a} (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

∫_{-a}^{a} (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* = 2*S*

= ∫_{0}^{a} (even) *dx*.

## Example 1

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

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

sin^{2} *θ* + cos^{2} *θ* = 1

Pythagorean identities

2 sin *θ* cos *θ* = 2 sin 2*θ*

sin 2*A* (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 2*θ*.

Solve the definite integral.

## Example 4

The interval of the integral is symmetric: [-*π*/4, *π*/4].

And tan *θ* is an odd function.

So this interval is 0.