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

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.

Example 6 :

Solution :

Let f(x) = sin²x ⋅ cosx.

f(-x) = sin²(-x) ⋅ cos(-x)

f(-x) = [sin(-x)]² ⋅ cosx

f(-x) = (sinx)² ⋅ cosx

f(-x) = sin²x ⋅ 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.

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.

Example 11 :

Solution :

f(-x) = -f(x)

f(x) is an odd function.

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