Problem 13 - Thomas Calculus

Question

In Exercises \(1-16,\) find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. $$\text { a. }\sec ^{2} x \quad \text { b. } \frac{2}{3} \sec ^{2} \frac{x}{3} \quad \text { c. }-\sec ^{2} \frac{3 x}{2}$$

Step-by-Step Solution

Answer

a. \( \tan x + C \), b. \( 2 \tan \left( \frac{x}{3} \right) + C \), c. \( -\frac{2}{3} \tan \left( \frac{3x}{2} \right) + C \).

Step 1: Understanding the Problem

We are given three functions for which we need to find antiderivatives. The process involves finding a function whose derivative is the given function. We will use standard integration techniques and known antiderivatives.

Step 1: Antiderivative of \( \sec^2 x \)

The function \( \sec^2 x \) is a standard derivative that corresponds to the tangent function. Hence, the antiderivative of \( \sec^2 x \) is \( \tan x + C \), where \( C \) is the constant of integration.

Step 2: Antiderivative of \( \frac{2}{3} \sec^2 \frac{x}{3} \)

To find the antiderivative of \( \frac{2}{3} \sec^2 \frac{x}{3} \), recognize this involves a chain rule in reverse. Let \( u = \frac{x}{3} \), then \( du = \frac{1}{3} dx \). Thus, the integral becomes:\[\int \frac{2}{3} \sec^2 \left( \frac{x}{3} \right) dx = 2 \int \sec^2(u) du = 2 \tan(u) + C = 2 \tan \left( \frac{x}{3} \right) + C .\]

Step 3: Antiderivative of \( -\sec^2 \frac{3x}{2} \)

For the function \( -\sec^2 \frac{3x}{2} \), again use a substitution: let \( v = \frac{3x}{2} \), so \( dv = \frac{3}{2} dx \), or \( dx = \frac{2}{3} dv \). The integral becomes:\[\int -\sec^2 \left( \frac{3x}{2} \right) dx = -\frac{2}{3} \int \sec^2(v) dv = -\frac{2}{3} \tan(v) + C \ = -\frac{2}{3} \tan \left( \frac{3x}{2} \right) + C .\]

Step 5: Verification by Differentiation

We can verify each result by differentiating them to see if we get the original function back. For example, differentiating \( \tan x \) yields \( \sec^2 x \), differentiating \( 2 \tan \left( \frac{x}{3} \right) \) yields \( \frac{2}{3} \sec^2 \frac{x}{3} \), and differentiating \( -\frac{2}{3} \tan \left( \frac{3x}{2} \right) \) yields \( -\sec^2 \left( \frac{3x}{2} \right) \). This confirms our antiderivatives are correct.

Key Concepts

Integration

Integration is a fundamental concept in calculus, akin to summing an infinite number of infinitesimally small quantities. When you integrate a function, you essentially find another function whose derivative gives you the original function. This process is called finding the antiderivative.

For instance, if you have a function like \(\sec^2 x\), and seek its antiderivative, integration leads you to \(\tan x + C\), where \(C\) represents the constant of integration. Integration uses known rules and formulas to deduce these antiderivatives easily.

Some essential points about integration include:

• Integration reverses differentiation.
• After integrating, always add a constant \(C\), since derivatives of constants are zero.
• Known functions, like \(\sec^2 x\), often have standard antiderivatives which simplify the integration process.

These principles are crucial when tackling problems where finding the antiderivatives may seem difficult at first glance. Understanding this core concept can make the process of integration straightforward and intuitive.

Differentiation

Differentiation is the flip side of integration and is a way to find the rate at which a function is changing at any given point. In simpler terms, it measures the slope or steepness of the tangent at a point on a curve. In exercises involving integration, differentiation is used as a verification tool.

Once you have found the antiderivative by integration, you differentiate it to check if it returns the original function. This serves as confirmation that the antiderivative was found correctly. For instance:

• Differentiating \(\tan x\) results in \(\sec^2 x\).
• Differentiating \(2 \tan\left(\frac{x}{3}\right)\) yields \(\frac{2}{3} \sec^2\left(\frac{x}{3}\right)\).
• Differentiating \(-\frac{2}{3} \tan\left(\frac{3x}{2}\right)\) provides \(-\sec^2\left(\frac{3x}{2}\right)\).

This step-by-step differentiation process helps substantiate your solutions obtained through integration, ensuring that your approach and techniques are sound.

Integration by Substitution

Integration by substitution simplifies complex integration tasks, often compared to the reverse of the chain rule in differentiation. It involves substituting a part of the integral with a new variable to simplify the integration process.

Consider the function \(\frac{2}{3} \sec^2 \frac{x}{3}\). Here, using substitution, let \(u = \frac{x}{3}\). This changes \(dx\) in terms of \(du\), transforming the integral into a more manageable form. After calculation, substitute back to get the integral in terms of the original variable.

Key steps in integration by substitution:

• Identify the inner function to substitute, making the integral easier.
• Express \(dx\) in terms of the new variable \(du\).
• Integrate with respect to \(du\) and substitute back to the original variable.

This method is invaluable for integrals that appear complex, allowing for a simplified process and often revealing clear and straightforward solutions.