Problem 45 - Biocalculus Calculus for the Life Sciences

Question

Evaluate the definite integral. \(\int_{-\pi / 4}^{\pi / 4}\left(x^{3}+x^{4} \tan x\right) d x\)

Step-by-Step Solution

Answer

The integral evaluates to 0.

Step 1: Separate the Integral

The given integral can be separated into two parts: \(\int_{-\pi/4}^{\pi/4} x^3 \, dx + \int_{-\pi/4}^{\pi/4} x^4 \tan x \, dx\). This allows us to evaluate each integral separately.

Step 2: Evaluate the First Integral

The first part is \(\int_{-\pi/4}^{\pi/4} x^3 \, dx\). This is an odd function integrated over a symmetric interval around zero, so the integral is zero. Thus, \(\int_{-\pi/4}^{\pi/4} x^3 \, dx = 0\).

Step 3: Evaluate Symmetry of the Second Function

Consider the function \(x^4 \tan x\). Note that \(x^4\) is an even function and \(\tan x\) is an odd function. The product of an even function and an odd function is an odd function. Thus, \(x^4 \tan x\) is odd.

Step 4: Apply Symmetry Property to the Second Integral

Since \(x^4 \tan x\) is odd and \(\int_{-a}^{a} f(x) \, dx = 0\) for any continuous odd function \(f(x)\), the integral \(\int_{-\pi/4}^{\pi/4} x^4 \tan x \, dx = 0\).

Step 5: Combine Results

Combining the results from the previous steps, \(\int_{-\pi/4}^{\pi/4} x^3 \, dx + \int_{-\pi/4}^{\pi/4} x^4 \tan x \, dx = 0 + 0 = 0\).

Key Concepts

Even and Odd Functions

In calculus, understanding even and odd functions is crucial for recognizing simple patterns that help solve integrals.
Even functions are symmetric around the y-axis, such as the function defined by the formula \(f(x) = x^2\). For these functions, \(f(x) = f(-x)\).
Odd functions are symmetric around the origin like \(f(x) = x^3\), where \(f(x) = -f(-x)\).
These properties significantly influence the evaluation of integrals, especially in symmetric intervals.

• When you integrate an even function over a symmetric interval \([-a, a]\), you can evaluate the definite integral with ease, usually without zeroing out.
• An odd function integrated over a symmetric interval like \([-a, a]\) always results in zero. This property comes from the symmetry of these functions. The positive and negative parts of the function cancel each other out.

Integral Symmetry

Integral symmetry simplifies definite integrals significantly.
When dealing with symmetric intervals, such as \([-\pi/4, \pi/4]\), we utilize the properties of even and odd functions.
These properties help cancel out or simplify the computation of the integrals.
Take the two parts of the original integral:

• \(x^3\) is an odd function. When integrated over \([-\pi/4, \pi/4]\), due to its nature, it gives zero.
• \(x^4\) multiplied by \(\tan x\) results in an odd function (since the product of even and odd functions is odd). This also results in a definite integral of zero over a symmetric interval.

By recognizing and applying symmetry properties, one can determine the results of certain integrals easily without performing complex calculations.

Calculus Problem Solving

Solving calculus problems like definite integrals can seem daunting, but recognizing patterns or properties can make the process easier.
The first step is typically to separate complicated expressions into simpler parts. In the given integral, we split it into \(\int_{-\pi/4}^{\pi/4} x^3 \, dx\) and \(\int_{-\pi/4}^{\pi/4} x^4 \tan x \, dx\).
By decomposing the integral into parts, we can examine each one based on its individual properties:

• Use symmetry: Identify if parts of the integrals are even or odd functions and simplify the process by using symmetry properties.
• Simplify: Take advantage of these properties to avoid lengthy computations. Recognize when things cancel out.
• Combine results: After evaluating each part, combine results for the final solution.

Through these methods, one can navigate complex calculus problems by systematically applying function properties and simplifying calculations.