Problem 61 - Calculus: Early Transcendentals

Question

Evaluate the definite integral. \( \displaystyle \int^{\pi/4}_{-\pi/4} (x^3 + x^4 \tan x) \, dx \)

Step-by-Step Solution

Answer

The definite integral is 0.

Step 1: Break Down the Integral

First, notice that the integral is composed of two separate terms: \( x^3 \) and \( x^4 \tan x \). We will evaluate these integrals separately and then combine the results.

Step 2: Evaluate the Integral of the First Term \( \int x^3 \, dx \)

Let's calculate \( \int x^3 \, dx \) with the given limits. The antiderivative of \( x^3 \) is \( \frac{x^4}{4} \). Now we evaluate it from \(-\frac{\pi}{4}\) to \(\frac{\pi}{4}\): \[ \left[ \frac{x^4}{4} \right]_{-\frac{\pi}{4}}^{\frac{\pi}{4}} = \frac{1}{4} \left[ \left(\frac{\pi}{4}\right)^4 - \left(-\frac{\pi}{4}\right)^4 \right] \]Both terms are equal, thus resulting in zero.

Step 3: Evaluate the Integral of the Second Term \( \int x^4 \tan x \, dx \)

This integral is more complex, but notice that it is an odd function over the symmetric interval \(-\pi/4\) to \(\pi/4\). An odd function, \(f(x) = -f(-x)\), when integrated over a symmetric interval around zero, yields zero. Therefore, \[ \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} x^4 \tan x \, dx = 0 \].

Step 4: Combine Results from Both Integrals

Since both integrals evaluate to zero, their sum is also zero. Thus, the final definite integral is zero.

Key Concepts

Antiderivative

When solving an integral, finding the antiderivative is one of the initial steps. The antiderivative, essentially the reverse process of a derivative, is a function whose derivative gives back the original function. To illustrate, consider the function \( x^3 \).
Finding its antiderivative involves reversing the power rule used in differentiation. For \( x^3 \), the antiderivative is \( \frac{x^4}{4} \). This is because when we differentiate \( \frac{x^4}{4} \), we get back \( x^3 \).
This process is crucial in definite integrals, where you find antiderivatives and then evaluate them at the given limits. In our example, integrating \( x^3 \) from \(-\frac{\pi}{4}\) to \(\frac{\pi}{4}\) involves substituting these limits into its antiderivative and calculating the difference. Because of symmetry, we found that both parts equal and cancel out, resulting in zero.

Odd Function

A function is termed odd when it satisfies the property \( f(x) = -f(-x) \). This is a peculiar symmetry across the origin of the coordinate system.
This symmetry becomes particularly useful in calculus when analyzing integrals. For example, consider the function \( x^4 \tan x \). It can be proven to be an odd function, which implies that the area under its curve from a negative point to a positive equivalent, such as from \(-a\) to \(a\), is zero.
This characteristic is extremely convenient. In the original exercise, since \( x^4 \tan x \) is odd, the integral across the interval \( -\pi/4 \) to \( \pi/4 \) is zero by virtue of its symmetry. This eliminates the need to perform a potentially complicated integration.

Symmetric Interval

Symmetrical intervals are those that are centered around zero on the real number line, such as \(-a\) to \(a\). Using these intervals can simplify the evaluation of integrals, especially when dealing with odd or even functions.
For odd functions like \( x^4 \tan x \), when integrated over symmetric intervals, the result is usually zero. This occurs because the negative half of the graph precisely cancels out the positive half.
Recognizing symmetric intervals can save time and effort. Rather than calculating complex integral values, understanding the symmetry can provide immediate insights and solutions.

• Symmetry leads to simplifications.
• It's often easier to analyze symmetric functions.
• Recognition of this can lead to shortcuts in math problems.

So, in the initial exercise, observing the symmetric interval from \(-\pi/4\) to \(\pi/4\) was key to deducing that the integral of the odd part of the function was zero.