Problem 37 - Calculus Single Variable

Question

Evaluate the definite integrals. $$ \int_{0}^{\pi / 8} \tan (2 x) d x $$

Step-by-Step Solution

Answer

The value of the integral is \( \frac{1}{4} \ln 2 \).

Step 1: Determine the Antiderivative

The first step is to accurately find the antiderivative of \( \tan(2x) \). We can rewrite \( \tan(2x) \) as \( \frac{\sin(2x)}{\cos(2x)} \). To integrate \( \tan(2x) \), recognize that its derivative is \( -\ln |\cos(2x)| \), but due to the chain rule, we must also divide by the derivative of \( 2x \), which is 2, resulting in \( -\frac{1}{2} \ln |\cos(2x)| \). Therefore, the antiderivative is \( -\frac{1}{2} \ln |\cos(2x)| + C \).

Step 2: Set up the Definite Integral

Insert the antiderivative \( -\frac{1}{2} \ln |\cos(2x)| \) in the definite integral from 0 to \( \frac{\pi}{8} \). This sets up the expression to evaluate: \[ \left[ -\frac{1}{2} \ln |\cos(2x)| \right]_{0}^{\pi / 8}. \]

Step 3: Evaluate the Antiderivative at the Upper Limit

Plug \( \frac{\pi}{8} \) into the antiderivative. This gives: \[ -\frac{1}{2} \ln |\cos(\frac{\pi}{4})|. \]Since \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \), the expression becomes \[ -\frac{1}{2} \ln \left(\frac{\sqrt{2}}{2}\right). \]

Step 4: Evaluate the Antiderivative at the Lower Limit

Now plug 0 into the antiderivative, which simplifies to: \[ -\frac{1}{2} \ln |\cos(0)|. \]Since \( \cos(0) = 1 \), this further simplifies to \[ -\frac{1}{2} \ln(1) = 0. \]

Step 5: Calculate the Definite Integral's Value

Calculate the value of the integral by subtracting the result of Step 4 from Step 3: \[ -\frac{1}{2} \ln \left(\frac{\sqrt{2}}{2}\right) - 0 = -\frac{1}{2} \ln \left(\frac{1}{\sqrt{2}}\right) = \frac{1}{2} \ln \sqrt{2}. \]Since \( \ln \sqrt{2} = \frac{1}{2} \ln 2 \), the result simplifies to\[ \frac{1}{4} \ln 2. \]

Key Concepts

Antiderivative

An antiderivative is a fundamental concept in calculus, especially in integral calculus. It refers to a function whose derivative gives you the original function that you started with. Finding an antiderivative is essentially the reverse process of differentiation.
In this exercise, we need to find the antiderivative of \( \tan(2x) \), which can be a bit tricky. We start by rewriting \( \tan(2x) \) as \( \frac{\sin(2x)}{\cos(2x)} \). This helps us prepare for integration. We aim to transform the function into a form that is easier to integrate. Using a known derivative fact, we know that the derivative of \( \ln|\cos(2x)| \) is \( -\tan(2x) \) multiplied by the derivative of the inside function \( 2x \), which means we get an antiderivative of \(-\frac{1}{2} \ln|\cos(2x)| + C\). Here, \( C \) is the constant of integration, which is not needed when evaluating definite integrals.

Trigonometric Functions

Trigonometric functions like \( \sin(x) \) and \( \cos(x) \) are pivotal not only in trigonometry but also in calculus when performing integrations. In this problem, \( \tan(x) \), which can be expressed as \( \frac{\sin(x)}{\cos(x)} \), is the focus.
The trigonometric function \( \tan(2x) \) is embedded in the definite integral from 0 to \( \frac{\pi}{8} \). Recognizing trigonometric identities like \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \) is crucial for simplifying expressions during integration. When evaluating the definite integral, you substitute the limits into the antiderivative and then simplify using such identities. This allows conversion of complex expressions into easier forms, making calculations more straightforward. Mastery of these functions and identities is essential for success in calculus.

Integration Techniques

Integration techniques are approaches used to solve integrals, and sometimes involve transforming the integral into a more manageable form. In the case of our exercise, identifying that \( \tan(2x) \) can be rewritten as \( \frac{\sin(2x)}{\cos(2x)} \) is a core step.
For integration, recognizing that the function can relate to a basic derivative form helps simplify the process. Here, the natural logarithmic derivative \( \ln|\cos(2x)| \) is pivotal since its derivative relates directly to \( \tan(2x) \). Moreover, integration by substitution and recognizing patterns involve using basic integration rules and sometimes more complex strategies like partial fraction decomposition or trigonometric substitutions.

• Apply transformation of trig functions into easier-to-integrate forms.
• Use logarithmic differentiation as an insight.
• Ensure that limits of integration are properly evaluated to get the final result.

These techniques not only simplify the existing problem but serve as a template for approaching a wide array of integrals in calculus.