Problem 66 - Essential Calculus

Question

(a) find the indefinite integral in two different ways. (b) Use a graphing utility to graph the antiderivative (without the constant of integration) obtained by each method to show that the results differ only by a constant. (c) Verify analytically that the results differ only by a constant. $$ \int \sec ^{2} x \tan x d x $$

Step-by-Step Solution

Answer

The indefinite integral can be solved in two ways: either by direct integration to find the answer as \( \frac{\tan^3 x}{3} + C1 \), or by using integration by substitution to find the answer as \( \frac{\tan^2 x}{2} + C2 \). Both graphically and analytically, it can be verified that these two solutions differ only by a constant.

Step 1: Method 1: Direct Integration

First, multiply \( \sec^2 x \) and \( \tan x \) to get \( \int \tan x \sec^2 x dx \). This integral can be directly solved as \( \frac{\tan^3 x}{3} + C1 \) where \( C1 \) is the constant of integration.

Step 2: Method 2: Integration by Substitution

This method involves making a substitution. Notice that the derivative of \( \tan x \) is \( \sec^2 x \), a good sign that we can let \( u = \tan x \). Then \( du = \sec^2 x dx \). The integral then becomes \( \int u du \) which can be integrated to yield \( \frac{u^2}{2} = \frac{\tan^2 x}{2} + C2 \) where \( C2 \) is the constant of integration.

Step 3: Graphical Verification

Using a graphing utility, plot both \( \frac{\tan^3 x}{3} \) and \( \frac{\tan^2 x}{2} \). Although the two functions appears different, the difference in their Y-values for the same X-value (which can be verified with several points) will always be constant.

Step 4: Analytical Verification

To analytically verify that the two results differ only by a constant, subtract the second function from the first, i.e. \( \frac{\tan^3 x}{3} - \frac{\tan^2 x}{2} \). Simplifying this expression will yield a constant difference that verifies our analytical conclusion.

Key Concepts

Direct Integration

Direct integration is one of the simplest techniques used in indefinite integral calculus. It's the process of integrating a function directly without applying any complicated methods or substitutions. In our discussed exercise, we see direct integration in action when we multiply \( \sec^2 x \) with \( \tan x \) to obtain \( \int \tan x \sec^2 x \,dx \). The antiderivative of this function can be found by recognizing a pattern or formula that fits the integrand. The result, \( \frac{\tan^3 x}{3} + C1 \) where \( C1 \) represents the integration constant, is obtained by applying power rule for integration directly.

Understanding the direct integration method is vital because it helps students quickly identify situations where integrals can be solved swiftly without further manipulation. When faced with a new integrand, always check if it can be integrated directly before attempting other more complex methods.

Integration by Substitution

Integration by substitution, also known as 'u-substitution', is a technique that simplifies the integration process by changing the variable of integration to something that is easier to manage. It's similar to the algebraic method of substitution used for solving equations. In the example problem, we apply integration by substitution by letting \( u = \tan x \) and \( du = \sec^2 x \,dx \). This transforms the given integral into \( \int u \,du \), which is straightforward to integrate into \( \frac{u^2}{2} \), or, reverting back to the original variable, \( \frac{\tan^2 x}{2} + C2 \) with \( C2 \) being another constant of integration.

This method is extremely useful when the integral contains a function alongside its derivative, indicating that a substitution could simplify the problem. Practice identifying these patterns to become proficient in using this indispensable integration tool.

Graphing Antiderivatives

Graphing antiderivatives provides a visual understanding of the relationship between a function and its integral. It can confirm whether different methods of integration lead to the same family of antiderivatives differing only by a constant. The exercise requires the use of graphing software to plot the antiderivatives obtained by direct integration and by substitution: \( \frac{\tan^3 x}{3} \) and \( \frac{\tan^2 x}{2} \) respectively. Although they may not look the same, graphing confirms that they are vertically offset versions of each other, which corresponds with the idea that an indefinite integral can include an arbitrary constant.

Graphs can also provide insight into the behavior of antiderivatives, such as how they increase or decrease and their symmetry or periodicity. When learning integration, students should often refer to graphs to gain an intuitive grasp of the concept of antiderivatives.

Verification of Constants

Verification of constants in indefinite integrals is the analytical step to confirm that different methods yield results that differ only by a constant. After finding two expressions for the integral using different techniques, one must often prove that the two expressions represent the same antiderivative apart from a constant. In our example problem, we subtract one antiderivative from the other, \( \frac{\tan^3 x}{3} - \frac{\tan^2 x}{2} \). Simplifying this yields a constant, \( C1 - C2 \), thus analytically verifying our predicted relationship between the two solutions.

This verification step brings closure to the problem-solving process and ensures that all solutions are valid, regardless of the method used. It is a fundamental concept within integral calculus that aids in the understanding of the indefinite nature of integrals and the impact of constants on families of functions.