Problem 26 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \sec ^{2}(x+2) d x$$

Step-by-Step Solution

Answer

The result is \(\tan(x+2) + C\).

Step 1: Identify substitution

Identify a substitution that will simplify the integral. In this case, let \(u = x + 2\). The differential \(du\) is equal to \(dx\).

Step 2: Rewrite the Integral

Replace all instances of x with u in the integral. Also, replace dx with du. We then obtain \(\int \sec^{2}(u) du\).

Step 3: Evaluate the Integral

Now that the integral has been simplified, it can be evaluated. The integral of the secant squared function is the tangent function, so \(\int \sec^{2}(u) du = \tan(u) + C\).

Step 4: Substitute u back

Now, replace u with \(x + 2\) (from the initial substitution defined in step 1). Therefore, the result is \(\tan(x+2) + C\).

Key Concepts

Indefinite Integral

An indefinite integral, also known as an antiderivative, is a concept from calculus that represents a broad family of functions whose derivative would give the original function. In other words, when you're given a function such as \( f(x) \), finding the indefinite integral involves discovering a function \( F(x) \) such that \( F'(x) = f(x) \). Unlike definite integrals, indefinite integrals do not require bounds; instead, the result includes a constant of integration, typically denoted as \( C \), because the antiderivative is not unique.

The expression for an indefinite integral is written using the integral sign followed by the function and the differential operator, which, in the simplest form looks like \( \int f(x) \, dx \). This tells us to integrate the function \( f(x) \) with respect to \( x \). Improving your ability to evaluate indefinite integrals involves recognizing patterns in integrals, such as basic functions, and applying integration techniques appropriately.

Trigonometric Integration

Trigonometric integration deals with integrating functions that involve trigonometric functions like sine, cosine, tangent, and their reciprocals. These types of integrals are very common in calculus, especially in problems involving periodic functions or geometrical relationships.

For example, the integral in the textbook exercise we're looking at is \( \int \sec^2(x+2) dx \), which is a trigonometric integral involving the secant function squared. To integrate such functions, students must be familiar with the fundamental trigonometric identities, such as \( \frac{d}{dx} \tan(x) = \sec^2(x) \) and techniques like substitution to simplify the integral before evaluation. Understanding these identities is crucial because they often provide the straightforward antiderivatives needed to resolve the integral.

It's also helpful to remember the derivatives and integrals of basic trigonometric functions since these directly apply to many trigonometric integration problems. For instance, in our problem, recognizing that the antiderivative of \( \sec^2(x) \) is \( \tan(x) \) is key to solving the integral.

Integration Techniques

Integration techniques encompass the methods used to evaluate integrals that are not immediately straightforward. These methods include substitution (also known as u-substitution), integration by parts, partial fractions, trigonometric substitution, and others. The key is to transform the original integral into a form that is easier to evaluate or matches a known integral form.

Substitution, the method used in our example exercise, is often the first technique taught for tackling more complex integrals. It involves identifying a part of the integral (a function or an expression within the integral) that can be replaced with a single variable, making the integral simpler. To execute substitution:

• Choose a substitution that will simplify the integral (e.g., \( u = g(x) \)).
• Determine \( du \) and replace the differential in the integral (\( dx \)) with \( du \).
• Complete the integration with respect to the new variable.
• Substitute back to the original variable if necessary.

Properly applying these integration techniques often turns a challenging problem into a manageable calculation. As with any mathematical method, mastering these techniques requires practice and an understanding of when to apply each technique for efficient and effective problem-solving.