Question

Use symmetry to help you evaluate the given integral. $$ \int_{-\pi / 4}^{\pi / 4}\left(|x| \sin ^{5} x+|x|^{2} \tan x\right) d x $$

Short answer

The integral evaluates to 0.

Step-by-step solution

Analyze the Integral

We need to evaluate the integral \( \int_{-\pi / 4}^{\pi / 4} (|x| \sin^5 x + |x|^2 \tan x) \, dx \). Notice that it is composed of two parts: \( |x| \sin^5 x \) and \( |x|^2 \tan x \). Evaluate each part separately and check for symmetry.

Symmetry in the First Term

Examine \( |x| \sin^5 x \). Since \( |x| \) is an even function and \( \sin^5 x \) is an odd function (as \( (-x)^5 = -x^5 \)), the product \( |x| \sin^5 x \) will be an odd function. The integral of an odd function over a symmetric interval from \( -a \) to \( a \) is zero.

Symmetry in the Second Term

Next, consider \( |x|^2 \tan x \). The function \( |x|^2 \) is even, and \( \tan x \) is odd, which means their product \( |x|^2 \tan x \) is an odd function again. Hence, its integral over the symmetric interval \( -\pi/4 \) to \( \pi/4 \) is also zero.

Combine Results

Since both terms of the integral are odd functions, the integral of the entire expression \( |x| \sin^5 x + |x|^2 \tan x \) over the interval \( -\pi/4 \) to \( \pi/4 \) will be zero.

Key concepts

Symmetry in Integrals

Symmetry plays a crucial role in solving integrals, especially when dealing with definite integrals over symmetric limits. Symmetry is a property that helps to simplify calculations, saving time and effort by identifying certain integral characteristics.

When an integral is taken over a symmetric interval, like from \(-a\) to \(a\), the function's symmetry can make the integration process much simpler. Symmetric intervals mean that the interval has equal magnitude on both sides of zero. Often, functions exhibit symmetrical properties, and integrals over these can yield particularly simple results:

• If a function is even, i.e., symmetrical about the y-axis, then its behavior over the interval \([-a, a]\) can lead to simplifications or duplications in calculation.
• If a function is odd, meaning its graph is symmetric about the origin, the integral of an odd function over a symmetric interval is zero.

Using symmetry effectively allows us to assess which parts of an integral may sum to zero or simplify significantly, making it easier to evaluate complex integrals.

Odd and Even Functions

Understanding odd and even functions is essential in integral calculus, particularly when dealing with symmetrical limits.

An **even function** is a function where \(f(-x) = f(x)\) for all \(x\) in the function's domain. Examples of even functions include \(x^2\), cosine functions, and any function that's symmetrical about the y-axis. The integral of an even function over a symmetric interval \([-a, a]\) is usually found by analyzing the symmetry properties, possibly reducing calculations.

On the other hand, an **odd function** satisfies \(f(-x) = -f(x)\). A classic example of an odd function is \(x^3\), sine functions, or any function that has symmetry about the origin. Integrals of odd functions over symmetric intervals \([-a, a]\) turn out to be zero because the areas under the curve on either side of the y-axis cancel each other out.

In the given exercise, both terms, \(|x|\sin^5(x)\) and \(|x|^2\tan(x)\), were found to be odd. Thus, their integrals over the symmetric interval \(-\frac{\pi}{4}\) to \(\frac{\pi}{4}\) are zero.

Evaluation of Definite Integrals

The evaluation of definite integrals involves calculating the overall area under a curve given an interval. It builds on the idea of accumulated change, where the integral represents the total sum of infinitesimal changes over a given interval.

When evaluating definite integrals, there are essential steps to consider:

• Identify the limits of integration – in this case, from \(-\frac{\pi}{4}\) to \(\frac{\pi}{4}\).
• Determine the function within the integral. Analyze the function to decide if any properties, like symmetry, can simplify the integration process.
• Apply fundamental theorems of calculus where needed, such as using antiderivatives and evaluating at boundary limits for non-zero evaluations.

In the exercise provided, because both component functions \(|x|\sin^5(x)\) and \(|x|^2\tan(x)\) were identified as odd over a symmetric interval, the integral was found to be zero.

Effective evaluation methods can save time and avoid complex calculations, emphasizing the importance of recognizing function properties, such as odd or even functions, in definite integral evaluation.