Problems based on a+b-x property in Definite Integrals

The a+b-x property is a useful tool in the evaluation of definite integrals, particularly when dealing with symmetric limits. This property is based on the concept of even and odd functions and their behavior under integration over symmetric intervals.

Understanding the a+b-x Property

The a+b-x property states that for a definite integral with limits a and b, if the integrand can be expressed in terms of a+b-x, certain symmetries can be exploited to simplify the integral. This is particularly useful when the integrand is an even or odd function.

Even and Odd Functions

A function f(x) is called even if f(-x) = f(x) for all x in the domain of f. Graphically, even functions are symmetric about the y-axis.

A function f(x) is called odd if f(-x) = -f(x) for all x in the domain of f. Graphically, odd functions have rotational symmetry about the origin.

Application in Definite Integrals

When integrating an even function over a symmetric interval [-a, a], the integral can be simplified as:

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

For an odd function, the integral over a symmetric interval is always zero:

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

Using the a+b-x Property

When faced with an integral of the form:

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

We can sometimes make a substitution x = a+b-t to reveal the symmetry of the function. If the resulting function f(a+b-t) exhibits even or odd symmetry, we can apply the above simplifications.

Differences and Important Points

Property	Even Function	Odd Function
Definition	f(-x) = f(x)	f(-x) = -f(x)
Symmetry	Symmetric about y-axis	Symmetric about origin
Integral over symmetric interval	2\int_{0}^{a} f(x) \, dx	0
Example	f(x) = cos(x)	f(x) = sin(x)

Examples

Example 1: Even Function

Consider the integral:

$$ \int_{-2}^{2} x^2 \, dx $$

The function f(x) = x^2 is an even function. Therefore, we can simplify the integral as:

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

Example 2: Odd Function

Consider the integral:

$$ \int_{-3}^{3} x^3 \, dx $$

The function f(x) = x^3 is an odd function. Therefore, the integral evaluates to zero:

$$ \int_{-3}^{3} x^3 \, dx = 0 $$

Example 3: Using a+b-x Substitution

Evaluate the integral:

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

Let's use the substitution x = \pi - t. Then dx = -dt, and when x = 0, t = \pi, and when x = \pi, t = 0. The integral becomes:

$$ \int_{\pi}^{0} (\pi - t)(-\sin(t)) \, (-dt) $$

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

Now, we can split the integral into two parts:

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

The first part is an odd function integrated over a symmetric interval, so it is zero. The second part does not simplify directly, but the a+b-x property has helped us identify a part of the integral that is zero, simplifying our computation.

In conclusion, the a+b-x property is a powerful technique for simplifying definite integrals by exploiting the symmetry of the integrand. Understanding even and odd functions and their behavior under integration is crucial for effectively using this property.