Question

Perform the indicated integrations. $$ \int e^{x} \sec ^{2}\left(e^{x}\right) d x $$

Short answer

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

Step-by-step solution

Identify the Substitution

To simplify the integration, notice that the expression inside the secant squared is a function of an exponential term. We use substitution by letting \( u = e^{x} \). Then, the differential \( du = e^{x} dx \) becomes quite useful in simplifying the integral.

Substitute and Rewrite the Integral

Replace \( e^x \) with \( u \) in the integral. The original integral \( \int e^{x} \sec ^{2}(e^{x}) dx \) becomes \( \int \sec^2(u) du \), because \( e^x dx = du \).

Integrate Using Basic Trigonometric Integrals

The integral of \( \sec^2(u) \) with respect to \( u \) is known to be \( \tan(u) \). So the integral \( \int \sec^2(u) du \) is equal to \( \tan(u) + C \), where \( C \) is the constant of integration.

Back-Substitute to Original Variable

Replace \( u \) with its original expression in terms of \( x \), which is \( e^{x} \). Thus, \( \tan(u) + C \) becomes \( \tan(e^{x}) + C \).

Key concepts

Exponential Functions

Exponential functions are a key concept in integral calculus and represent expressions where a constant base is raised to a variable exponent. The most common base for exponential functions is the natural constant \( e \), approximately equal to 2.718.
These functions are characterized by their growth rate, which is proportional to their current value.
This particular property makes them pervasive in describing real-world phenomena such as population growth or radioactive decay.

When dealing with integrals involving exponential functions, recognizing how to utilize substitution can massively simplify the process.

• For instance, with an integral like \( \int e^{x} \sec ^{2}(e^{x}) dx \), identifying the inner term as an exponential helps in choosing the appropriate substitution.
• We set \( u = e^{x} \), which simplifies our calculations considerably.

This substitution allows us to focus on integrating simpler expressions and makes handling complex integration a bit more intuitive.

Trigonometric Substitution

Trigonometric substitution is a powerful technique used in integration to handle integrals involving trigonometric expressions like \( \sec(x), \sin(x), \tan(x) \), and others.
It often involves replacing a complex trigonometric expression with a simpler one, aiding in the integration process.
In some cases, this might also mean transforming an integral into one involving a trigonometric function from an algebraic one.

For the problem \( \int e^{x} \sec^{2}(e^{x}) dx \), trigonometric substitution simplifies the function \( \sec^{2}(e^{x}) \).

• After setting \( u = e^{x} \), we observe that the problem reduces to a standard trigonometric integral \( \int \sec^2(u) du \).
• The integral of \( \sec^2(u) \) is directly known as \( \tan(u) \), echoing the derivative of tangent functions.

This reduction is a classic example of how trigonometric substitution can transform a seemingly complex integral into a straightforward one.

Integration Techniques

Mastering integration techniques is crucial for solving integrals efficiently, especially when dealing with complex functions. Various methods, like substitution, partial fractions, integration by parts, and trigonometric substitution, provide diverse tools for approaching different problems.

In the context of \( \int e^{x} \sec^{2}(e^{x}) dx \), the chosen technique was substitution, which aligns with the structure of the integral.

• By selecting \( u = e^{x} \), the integral simplifies significantly, making \( \int \sec^2(u) du \) easily identifiable.
• This highlights the importance of recognizing patterns and choosing the right technique for the task.

Once simplified through substitution, the next step involves using basic integration rules.
Keen comprehension of these rules and their applications can turn even the most daunting integrals into comprehensible tasks.
Always remember, practice makes perfect when learning integration techniques.