Problem 26 - Calculus

Question

Evaluate \(\int_{-100}^{100}\left(x^{3}+\sin ^{5} x\right) d x\)

Step-by-Step Solution

Answer

The integral evaluates to 0.

Step 1: Break Down the Integral

First, separate the integral into two different integrals: \( \int_{-100}^{100} x^3\, dx \) and \( \int_{-100}^{100} \sin^5 x \; dx \). This is possible because the integral of a sum is the sum of the integrals.

Step 2: Evaluate the Symmetry of Each Function

The integrand \( \sin^5 x \) is an odd function because it is a sine function raised to an odd power, and \( x^3 \) is also an odd function. Integrals of odd functions over symmetric intervals around the origin, such as \([-a, a]\), evaluate to zero.

Step 3: Apply Symmetry Property

For the integral \( \int_{-100}^{100} x^3\, dx \) and \( \int_{-100}^{100} \sin^5 x \; dx \), since both \(x^3\) and \(\sin^5 x\) are odd functions over the interval \([-100, 100]\), both integrals equal zero.

Step 4: Combine the Results

Since both integrals evaluated to zero, the original integral \( \int_{-100}^{100} \left(x^3 + \sin^5 x\right) dx \) is the sum of the two, which is \(0 + 0 = 0\).

Key Concepts

Odd Functions

In mathematics, an odd function is a function that satisfies the condition \( f(-x) = -f(x) \) for all \( x \) in the function's domain. This means that if you flip the function around the y-axis and then around the x-axis, it looks the same. Graphically, odd functions have rotational symmetry around the origin, or they look the same if rotated 180 degrees.

Some common examples of odd functions include:

• \( x^3 \)
• \( \sin(x) \)
• \( x^5 \)

In the context of our exercise, both \( x^3 \) and \( \sin^5(x) \) are odd functions. The property of being odd is crucial when evaluating integrals over symmetric intervals.

Symmetry Property

The symmetry property of integrals is a valuable tool in integral calculus, especially when dealing with odd functions. This property refers to the effects of function symmetry on the outcome of definite integrals. If a function is odd,

• and it is integrated over an interval that is symmetric about the origin, \([-a, a]\),

then the integral of the function over this interval is zero.

This is because the areas above and below the x-axis cancel each other out. In our example, both \( x^3 \) and \( \sin^5(x) \) are integrated over the interval \([-100, 100]\), which is symmetrical.

Here are some points to note about symmetry:

• Even functions are symmetrical about the y-axis, and \( f(x) = f(-x) \).
• For even functions, the integral over a symmetric interval does not necessarily result in zero.

Understanding symmetry can simplify complex calculations significantly and is a cornerstone in definite integral evaluations.

Definite Integrals

A definite integral is a type of integral that calculates the "net area" under a curve, between two specified points. The notation \( \int_a^b f(x) \, dx \) is used to represent a definite integral of the function \( f(x) \) from point \( a \) to point \( b \).

Definite integrals have numerous applications, such as calculating areas, volumes, and even probabilities in more advanced settings. When integrals have limits that are the same (as with \([-a, a]\) in our problem), this symmetry can be used to simplify calculations.

In working with definite integrals, particularly in the context of odd functions:

• The results can often be obtained quickly by recognizing symmetry properties, as noted above.
• The integral of an odd function over a symmetric interval \([-a, a]\) will always be zero.
• You can return to fundamental definitions or graphical interpretations if ever in doubt about the problem's setup.

Definite integrals are fundamental not just in calculus, but across mathematics disciplines, making them vital for students to master.