# DEFINITE INTEGRALS OF ODD AND EVEN FUNCTIONS

In a function f(x),

f(-x) = -f(x) ----> f(x) is an odd function

f(-x) = f(x) ----> f(x) is an even function

## Definte Integrals of Odd and Even Functions

If f(x) is an odd function,

If f(x) is an even function,

Evaluate each of the following integrals.

Example 1 :

Solution :

Let f(x) = x3 + 3x.

f(-x) = (-x)3 + 3(-x)

f(-x) = -x3 - 3x

f(-x) = -(x3 + 3x)

f(-x) = -f(x)

f(x) is an odd function.

Example 2 :

Solution :

Let f(x) = 3x2 + 2.

f(-x) = 3(-x)2 + 2

f(-x) = 3x2 + 2

f(-x) = f(x)

f(x) is an even function.

Example 3 :

Solution :

Let f(x) = sinx ⋅ cos4x.

f(-x) = sin(-x) ⋅ cos4(-x)

f(-x) = -sinx ⋅ [cos(-x)]4

f(-x) = -sinx ⋅ (cosx)4

f(-x) = -sinx ⋅ cos4x

f(-x) = -f(x)

f(x) is an odd function.

Example 4 :

Solution :

Let f(x) = x3 ⋅ cos3x.

f(-x) = (-x)3 ⋅ cos3(-x)

f(-x) = -x3 ⋅ [cos(-x)]3

f(-x) = -x3 ⋅ (cosx)3

f(-x) = -x3 ⋅ cos3x

f(-x) = -f(x)

f(x) is an odd function.

Example 5 :

Solution :

Let f(x) = cos3x.

f(-x) = cos3(-x)

f(-x) = [cos(-x)]3

f(-x) = (cosx)3

f(-x) = cos3x

f(-x) = f(x)

f(x) is an even function.

Cosine Triple Angle Identity :

cos3x = 4cos3x - 3cosx

Example 6 :

Solution :

Let f(x) = sin2⋅ cosx.

f(-x) = sin2(-x) ⋅ cos(-x)

f(-x) = [sin(-x)]2 ⋅ cosx

f(-x) = (sinx)2 ⋅ cosx

f(-x) = sin2⋅ cosx

f(-x) = f(x)

f(x) is an even function.

Example 7 :

Solution :

Let f(x) = x ⋅ sin2x.

f(-x) = (-x) sin2(-x)

f(-x) = -x ⋅ [sin(-x)]2

f(-x) = -x ⋅ (-sinx)2

f(-x) = -x ⋅ sin2x

f(-x) = -f(x)

f(x) is an odd function.

Example 8 :

Solution :

Let f(x) = x3 ⋅ sin2x.

f(-x) = (-x)3 sin2(-x)

f(-x) = -x3 ⋅ [sin(-x)]2

f(-x) = -x3 ⋅ (-sinx)2

f(-x) = -x3 ⋅ sin2x

f(-x) = -f(x)

f(x) is an odd function.

Example 9 :

Solution :

Let f(x) = sin2x.

f(-x) = sin2(-x)

f(-x) = [sin(-x)]2

f(-x) = (-sinx)2

f(-x) = sin2x

f(-x) = f(x)

f(x) is an even function.

Cosine Double Angle Identity :

cos2x = cos2x - sin2x

cos2x = 1 - sin2x - sin2x

cos2x = 1 - 2sin2x

2sin2x = 1 - cos2x

Example 10 :

Solution :

Let f(x) = x ⋅ sinx.

f(-x) = (-x) ⋅ sin(-x)

f(-x) = -x ⋅ (-sinx)

f(-x) = x ⋅ sinx

f(-x) = f(x)

f(x) is an even function.

Integration by Parts Formula :

Let u = x and dv = sinxdx.

 u = xᵈᵘ⁄dₓ = 1du = dx

Example 11 :

Solution :

f(-x) = -f(x)

f(x) is an odd function.

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