Problem 30 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-54, perform the indicated integrations. \(\int e^{x} \sec ^{2}\left(e^{x}\right) d x\)

Step-by-Step Solution

Answer

The integral is \( \tan(e^x) + C \).

Step 1: Identify the Integration Technique

To solve the integral \( \int e^{x} \sec^{2}(e^{x}) \, dx \), we recognize that this is an integrand where substitution will be beneficial. Notice the function \( e^x \) inside \( \sec^2(e^x) \) and in the differential \( e^x \, dx \). This suggests using a substitution method.

Step 2: Apply Substitution

Let \( u = e^x \). Then, the differential \( du = e^x \, dx \). This substitution simplifies the integral in terms of \( u \). Substituting, the integral becomes \( \int \sec^2(u) \, du \).

Step 3: Integrate with Respect to u

The integral \( \int \sec^2(u) \, du \) is a standard result, where \( \int \sec^2(u) \, du = \tan(u) + C \). Compute the integral to get \( \tan(u) + C \).

Step 4: Substitute Back to Original Variable

Since \( u = e^x \), substitute back to express the answer in terms of \( x \): \( \tan(e^x) + C \). Hence, the final solution to the integral \( \int e^{x} \sec ^{2}(e^{x}) \, dx \) is \( \tan(e^x) + C \).

Key Concepts

Integration Techniques

Integration by substitution is a powerful tool in the realm of calculus, especially when tackling integrals that contain a function and its derivative. This technique involves simplifying a complex integral into a more manageable form by making a suitable substitution. When using substitution:

• Identify the inner function in your integrand that, along with its derivative, can be isolated. This function is often the one inside another function or is multiplied by its derivative itself.
• Replace this function with a new variable, usually denoted as \( u \). You'll also need to calculate its differential \( du \), which involves taking the derivative of the \( u \) function and converting the remaining terms in terms of \( u \).
• Rewrite the original integral in terms of \( u \). Hopefully, this new integral is something that is standard or easier to integrate.

By converting the integrand, integration by substitution simplifies the integration process and often reveals the antiderivative directly. It can resolve cases with products of functions or composite functions with their derivatives easily.

Indefinite Integrals

Indefinite integrals, unlike definite integrals, do not have upper and lower limits. They represent a family of functions and include an arbitrary constant, \( C \). The integral sign \( \int \) indicates that we need to find the antiderivative of the function given. For indefinite integrals:

• The result is a set of functions, all differing by a constant—hence the inclusion of the constant \( C \) in the solution.
• They provide the most general form of antiderivatives for a function. This general solution can be particularized if initial conditions or extra information is given.

Indefinite integrals allow us to reverse the process of differentiation. It's important to recognize standard integral formulas, such as \( \int \sec^2(u) \, du = \tan(u) + C \), used frequently when integrating trigonometric identities. These formulas serve as shortcuts in calculus problem solving, aiding us in finding antiderivatives quickly.

Calculus Problem Solving

Solving calculus problems often requires an understanding of various techniques, like integration by substitution, and recognition of integral types, such as indefinite integrals. In the problem at hand, you start by identifying the type of integral and the appropriate method to solve it. This requires:

• Analyzing the structure of the integrand—does it feature composite functions, or could it benefit from substitution?
• Utilizing transformations, like setting \( u = e^x \), thereby introducing a new variable to make the integral easier.
• Calculating necessary derivatives and rewriting the integral with respect to the new variable to simplify the process.
• Finally, integrating with respect to the new variable and substituting back to express the solution in the original variable.

Calculus problem solving is all about strategy and careful step-by-step execution. Each technique acts like a tool in your toolbox, allowing you to unravel complex integrals with ease and precision. The process as a whole enhances analytical skills and deepens comprehension of mathematical concepts.