Problem 36 - Calculus

Question

Use symmetry to help you evaluate the given integral. $$ \int_{-1}^{1} \frac{x^{3}}{\left(1+x^{2}\right)^{4}} d x $$

Step-by-Step Solution

Answer

The integral is 0 because the integrand is an odd function over a symmetric interval.

Step 1: Identify Symmetry

Recognize the symmetry of the function inside the integral. The function \( f(x) = \frac{x^3}{(1+x^2)^4} \) is an odd function because \( f(-x) = -f(x) \). This symmetry can simplify the integration process.

Step 2: Evaluate the Symmetry Property of Definite Integrals

Understand that an integral of an odd function over a symmetric interval around zero, such as \([-a, a]\), is zero. This is because the areas under the curve on either side of the y-axis cancel each other out.

Step 3: Conclusion

Since the function \( f(x) = \frac{x^3}{(1+x^2)^4} \) is odd and integrated over the symmetric interval \([-1, 1]\), the integral evaluates to zero.

Key Concepts

Symmetry in Integrals

In calculus, symmetry can be a powerful tool when evaluating integrals. When we talk about symmetry in the context of integrals, we’re typically referring to the symmetry of the function being integrated. This means we're interested in how the function behaves on either side of the y-axis. If a function is symmetric about the y-axis or the origin, this can give us clues about its behavior over certain intervals.

In the case of definite integrals, if a function exhibits symmetry, we can sometimes simplify the calculation significantly, or even evaluate the integral with minimal computation. This is especially true for functions that are either

• even (symmetric about the y-axis) or
• odd (symmetric about the origin).

Recognizing these symmetries can save you from lengthy calculations and help you arrive at results more quickly.

Odd Functions

Odd functions are fascinating mathematical entities with a neat property: they offer a certain symmetry around the origin. Mathematically, a function is considered odd if for every input \( x \), the function satisfies \( f(-x) = -f(x) \). This tells us that the graph of the function is a mirror image on either side of the origin, but inverted.

This oddness leads us to an interesting result when calculating definite integrals: if an odd function is integrated over an interval that is symmetric around zero (like [-a, a]), the integral evaluates to zero. This happens because the areas under the curve on the negative side of the y-axis perfectly cancel out the areas on the positive side.

Identifying whether a function is odd is crucial in simplifying calculations, as it allows the immediate conclusion that the integral over the symmetric interval around zero is zero, without further integration work.

Definite Integrals

Definite integrals are a fundamental concept in calculus. They represent the area under a curve over a specified interval. For a given function \( f(x) \), the definite integral evaluates the net area between the function and the x-axis from the lower limit \( a \) to the upper limit \( b \).

One of the essential properties of definite integrals concerns symmetry, particularly in symmetric intervals like [-a, a]. For any function, if we can establish the type of symmetry (odd or even), it dramatically alters the simplicity of solving an integral. For odd functions, as discussed, the value of the definite integral over a symmetric interval is zero.

This property is an incredibly useful tool in calculus. It highlights not only the elegance of mathematical symmetry but also the practicality when dealing with complex functions. By recognizing symmetry early, we can avoid unnecessary calculations and understand the behavior of integrals more deeply and efficiently.