Understanding the "a-x" Property in Definite Integrals

The "a-x" property in definite integrals is a useful symmetry property that can simplify the evaluation of certain integrals. It is particularly helpful when dealing with functions that are even or odd with respect to a particular point, usually the midpoint of the interval of integration.

The "a-x" Property

The "a-x" property is based on the substitution $x = a - t$ within the definite integral. This substitution can sometimes simplify the integral or allow us to recognize symmetry that leads to an easier computation.

Table of Differences and Important Points

Property	Description	Example
Even Function	$f(a-x) = f(x)$	$f(x) = \cos(x)$
Odd Function	$f(a-x) = -f(x)$	$f(x) = \sin(x)$
Symmetry about $x=a/2$	Integral from $0$ to $a$ can be simplified	$\int_0^a f(x)\,dx = 2\int_0^{a/2} f(x)\,dx$ if $f(x)$ is even
Symmetry about $x=a/2$	Integral from $0$ to $a$ is zero	$\int_0^a f(x)\,dx = 0$ if $f(x)$ is odd

Formulas

For an even function $f(x)$, the following formula applies:

$$ \int_{0}^{a} f(x)\,dx = 2\int_{0}^{a/2} f(x)\,dx $$

For an odd function $f(x)$, the following formula applies:

$$ \int_{0}^{a} f(x)\,dx = 0 $$

Examples

Example 1: Even Function

Consider the function $f(x) = \cos(x)$, which is an even function. Let's evaluate the integral from $0$ to $\pi$:

$$ \int_{0}^{\pi} \cos(x)\,dx $$

Using the "a-x" property, we can rewrite the integral as:

$$ \int_{0}^{\pi} \cos(\pi - x)\,dx = \int_{0}^{\pi} \cos(x)\,dx $$

Since $\cos(x)$ is even, we can use the formula for even functions:

$$ \int_{0}^{\pi} \cos(x)\,dx = 2\int_{0}^{\pi/2} \cos(x)\,dx $$

Example 2: Odd Function

Consider the function $f(x) = \sin(x)$, which is an odd function. Let's evaluate the integral from $0$ to $\pi$:

$$ \int_{0}^{\pi} \sin(x)\,dx $$

Using the "a-x" property, we can rewrite the integral as:

$$ \int_{0}^{\pi} \sin(\pi - x)\,dx = -\int_{0}^{\pi} \sin(x)\,dx $$

Since $\sin(x)$ is odd, the integral over the symmetric interval from $0$ to $\pi$ is zero:

$$ \int_{0}^{\pi} \sin(x)\,dx = 0 $$

Example 3: Using "a-x" Substitution

Let's evaluate the integral:

$$ \int_{0}^{2} (x^2 - 4x + 4)\,dx $$

We notice that the function is a perfect square and can be rewritten as:

$$ \int_{0}^{2} (x - 2)^2\,dx $$

Using the substitution $x = 2 - t$, we get:

$$ \int_{0}^{2} (2 - t)^2\,(-dt) = -\int_{2}^{0} t^2\,dt = \int_{0}^{2} t^2\,dt $$

Now, we can easily evaluate the integral:

$$ \int_{0}^{2} t^2\,dt = \left[\frac{t^3}{3}\right]_{0}^{2} = \frac{8}{3} $$

In conclusion, the "a-x" property is a powerful tool for simplifying definite integrals, especially when dealing with functions that exhibit symmetry. Recognizing even and odd functions and applying the appropriate formulas can greatly reduce the complexity of integral calculations.