Problem 51 - Calculus for Biology and Medicine

Question

Use substitution to evaluate the definite integrals. $$ \int_{0}^{\pi / 4} \tan x \sec ^{2} x d x $$

Step-by-Step Solution

Answer

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

Step 1: Identify Substitution Variable

To simplify the integral, we begin by identifying a substitution variable. Notice that the derivative of \(\tan x\) is \(\sec^2 x\). So, we set \(u = \tan x\).

Step 2: Compute Differential

Calculate the differential \(du\) in terms of \(dx\). Since \(\frac{d}{dx}(\tan x) = \sec^2 x\), we have \(du = \sec^2 x \, dx\).

Step 3: Change Limits of Integration

With \(u = \tan x\), we need to change the limits of integration accordingly. When \(x = 0\), \(u = \tan(0) = 0\). When \(x = \pi/4\), \(u = \tan(\pi/4) = 1\). Therefore, the limits in terms of \(u\) are from 0 to 1.

Step 4: Substitute Variables in the Integral

Substitute \(u = \tan x\) and \(du = \sec^2 x \, dx\) into the integral. The expression becomes \(\int_{0}^{1} u \, du\).

Step 5: Evaluate the Resulting Integral

Evaluate the integral \(\int_{0}^{1} u \, du\). The antiderivative of \(u\) is \(\frac{1}{2}u^2\), so the integral becomes \(\left[ \frac{1}{2}u^2 \right]_{0}^{1}\).

Step 6: Calculate the Definite Integral

Substitute the limits of integration into \(\frac{1}{2}u^2\): \(\frac{1}{2}(1)^2 - \frac{1}{2}(0)^2 = \frac{1}{2} - 0 = \frac{1}{2}\).

Key Concepts

Understanding the Substitution Method

The substitution method is a fundamental technique in calculus used to simplify complex integrals, particularly those involving products of functions and their derivatives. This process involves selecting a substitution variable, typically denoted as \(u\), that simplifies the integral's evaluation. In the given exercise, we chose \(u = \tan x\) because its derivative, \(\frac{d}{dx}(\tan x) = \sec^2 x\), appears in the integrand. This alignment between the substitution and the derivative makes it easier to rewrite the integral in terms of \(u\), simplifying the problem.

• This technique transforms a potentially intricate integral into a more manageable format.
• Effective when the integral contains a function and its derivative.
• Helps avoid complex mathematical manipulation that can lead to errors.

Once the substitution is made, it's important to express \(dx\) in terms of \(du\). With \(u = \tan x\), we find \(du = \sec^2 x \, dx\), allowing us to replace \(\sec^2 x \, dx\) in the original integral with \(du\). This conversion is crucial for simplifying the integral before evaluation.

Effective Calculus Problem Solving

Calculus problem solving often requires a strategic approach, combining different techniques to find a solution. In this exercise, the clear identification of the substitution method demonstrates this principle. By accurately pinpointing \(u = \tan x\) and understanding the relationship \(du = \sec^2 x \, dx\), we laid the groundwork for converting the integral into a simpler form, \(\int u \, du\). Here are some key strategies:

• Identify patterns that can simplify the integral, such as recognizing derivatives.
• Change variables mindfully, considering how \(u\) relates to \(x\). This ensures the problem remains manageable and consistent.
• Think ahead about how each step affects the integration limits or the overall problem structure.

These strategies are indispensable for tackling calculus problems, guiding you through a logical sequence of steps while maintaining a robust mathematical foundation. It's not just about performing operations but understanding the "why" behind each step.

Adjusting Integration Limits

After choosing a substitution variable, the next critical step is changing the integration limits. These limits correspond to the values \(u\) takes when \(x\) varies from its original limits. In our exercise, as \(x\) changes from 0 to \(\pi/4\), \(u\) changes from \(\tan(0) = 0\) to \(\tan(\pi/4) = 1\). Adjusting the limits ensures that the definite integral remains accurate in terms of the new variable, \(u\).

• This transformation maintains the integrity of the original integral.
• Ensures the evaluation produces the correct value for the definite integral.
• Without proper adjustment, there is a risk of incorrect results.

Remember, the new limits are always derived by substituting into the equation \(u = \tan x\). Once the new integration limits are set, the evaluation of the integral becomes straightforward. This subtle but crucial step ensures your solution is both valid and synonymous with the original problem's boundaries.