Problem 16 - Advanced Engineering Mathematics

Question

$$\int_{-\infty}^{\infty} \frac{x}{\left(x^{2}+4\right)^{3}} d x=0 \quad(\text { The integrand is an odd function })$$

Step-by-Step Solution

Answer

The integral is zero because the integrand is an odd function over a symmetric interval.

Step 1: Identify the Function

The integrand we have is \( f(x) = \frac{x}{(x^2+4)^3} \). We need to determine if this function is odd.

Step 2: Define an Odd Function

A function \( f(x) \) is odd if for all \( x \), \( f(-x) = -f(x) \). We will apply this definition to our integrand.

Step 3: Apply the Definition

Compute \( f(-x) = \frac{-x}{((-x)^2 + 4)^3} = \frac{-x}{(x^2 + 4)^3} = -\frac{x}{(x^2 + 4)^3} = -f(x) \). This shows that \( f(x) \) is indeed an odd function.

Step 4: Properties of Definite Integrals of Odd Functions

It is a known property that the integral of an odd function over a symmetric interval \( [-a, a] \) is zero. Hence, the integral from \( -\infty \) to \( \infty \) of an odd function is zero.

Step 5: Calculate the Integral

Since the integrand \( \frac{x}{(x^2+4)^3} \) is an odd function, we apply the property: \[ \int_{-\infty}^{\infty} \frac{x}{(x^2+4)^3} \, dx = 0. \]

Key Concepts

Odd Functions

Understanding odd functions helps us simplify many mathematical tasks. An odd function, by definition, satisfies the condition \( f(-x) = -f(x) \) for all \( x \). This means that if you take the function value at a negative \( x \) and it equals the negative of the function value at positive \( x \), then the function is odd.
This property results in odd functions having a symmetrical shape about the origin.In the exercise, the integrand \( f(x) = \frac{x}{(x^2+4)^3} \) is identified as odd because it meets the odd function criterion.
To verify, replace \( x \) with \(-x\), giving \( f(-x) = \frac{-x}{(x^2+4)^3} = -f(x) \). This verification confirms the function's odd nature, aiding in evaluating definite integrals over symmetric limits.

Properties of Integrals

The understanding of integral properties can ease complex calculations. One such property relates to the integral of odd functions.
Specifically, if the function is odd, the integral of this function over a symmetric interval is zero. This can be mathematically represented as:

• \( \int_{-a}^{a} f(x) \, dx = 0 \)

Applying this rule to the exercise means that the integration of the odd function from \(-\infty\) to \(\infty\) yields zero.
This property is especially useful in cases where directly integrating is challenging or computationally intense, saving time and simplifying the process.

Symmetric Intervals

Symmetric intervals significantly impact how we evaluate integrals, particularly for odd functions. A symmetric interval is simply a range that is the same distance from a central point, usually the origin, extending equally in both the positive and negative directions.For functions like \( f(x) = \frac{x}{(x^2+4)^3} \), symmetric intervals such as \([-a, a]\) are crucial.
Due to the symmetry, the areas under the curve from \(-a\) to 0 and 0 to \(a\) will cancel each other out if the function is odd. This cancellation leads directly to the integral being zero, as is the case in our exercise.
This concept thus simplifies analyzing and solving integrals by decreasing the need for complex calculations when conditions are met.