Problem 108 - Calculus for Biology and Medicine

Question

Evaluate the definite integrals. $$ \int_{-\pi / 4}^{\pi / 4} \tan x d x $$

Step-by-Step Solution

Answer

The definite integral is zero: \( \int_{-\pi / 4}^{\pi / 4} \tan x \, dx = 0 \).

Step 1: Identify the Function to Integrate

The function we need to integrate is \( \tan x \). We are going to evaluate the definite integral \( \int_{-\pi / 4}^{\pi / 4} \tan x \, dx \).

Step 2: Recognize the Antiderivative

The antiderivative of \( \tan x \) is \( -\ln|\cos x| \). This comes from the fact that \( \tan x = \frac{\sin x}{\cos x} \) and the derivative of \( \ln|\cos x| \) with respect to \( x \) is \( \frac{\sin x}{\cos x} \).

Step 3: Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus tells us to evaluate the antiderivative at the bounds. Therefore, we compute:\[-\ln|\cos(\pi/4)| - (-\ln|\cos(-\pi/4)|) = -\ln\left(\frac{\sqrt{2}}{2}\right) - (-\ln\left(\frac{\sqrt{2}}{2}\right)) = 0.\]

Step 4: Simplify the Expression

Given that the expressions \( -\ln\left(\frac{\sqrt{2}}{2}\right) \) and \( -(-\ln\left(\frac{\sqrt{2}}{2}\right)) \) are equal, it shows that the integral over the symmetric interval is zero due to the odd nature of \( \tan x \) around zero.

Key Concepts

Antiderivatives

When solving integrals, we often start by finding an antiderivative of the function we are integrating. An antiderivative, also known as an indefinite integral, is a function whose derivative is the original function given to us. For example, if we need to integrate \( \tan x \), we first need an antiderivative of \( \tan x \).
The antiderivative of \( \tan x \) is \( -\ln|\cos x| \). This result stems from using differentiation rules: if we take the derivative of \( \ln|\cos x| \), we obtain \( \tan x \). This reverse process of finding the original function from its derivative is key in definite integration.
Understanding antiderivatives is crucial as they allow us to compute the net area under a curve, between two points. Once we have the antiderivative, further steps such as applying the fundamental theorem of calculus become manageable.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) establishes a connection between differentiation and integration. It enables us to evaluate definite integrals efficiently using antiderivatives. There are two main parts to the theorem:

• The first part of the FTC tells us that an antiderivative can be used to evaluate the area under a curve.
• The second part specifies that given an antiderivative \( F(x) \), we evaluate the integral from \( a \) to \( b \) by computing \( F(b) - F(a) \).

For our specific integral \( \int_{-\pi / 4}^{\pi / 4} \tan x \, dx \), using the antiderivative \( -\ln|\cos x| \), we compute the difference: \[ -\ln|\cos(\pi/4)| - (-\ln|\cos(-\pi/4)|) = 0. \]
This step demonstrates how the FTC simplifies the process of finding net areas, turning a complex problem into manageable calculations.

Odd Functions

Odd functions exhibit symmetry characteristics that can simplify beyond just computation of integrals. A function \( f(x) \) is odd if it satisfies \( f(-x) = -f(x) \). These functions are symmetric about the origin.

• Tangent function \( \tan x \) is an example of an odd function.
• For an odd function with limits that are inverses, the integral over a symmetric interval around zero is always zero.

In our evaluated integral \( \int_{-\pi / 4}^{\pi / 4} \tan x \, dx \), the result was zero. This is because the function's positive and negative areas between symmetric limits cancel each other out.
Identifying odd functions early on in problems can save time and computational efforts by leveraging these symmetry properties.