Problem 339 - Calculus Volume 1

Question

In the following exercises, find each indefinite integral by using appropriate substitutions. $$\int e^{\tan x} \sec ^{2} x d x$$

Step-by-Step Solution

Answer

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

Step 1: Identify the substitution

We need to look for a substitution to simplify the integral \( \int e^{\tan x} \sec^{2} x \, dx \). Notice that the derivative of \( \tan x \) is \( \sec^{2} x \). This suggests we can use \( u = \tan x \) for substitution.

Step 2: Differentiate u

Differentiate \( u = \tan x \) with respect to \( x \): \[ \frac{du}{dx} = \sec^2 x \]Therefore, \( du = \sec^2 x \, dx \).

Step 3: Substitute variables

Substitute \( u = \tan x \) and \( du = \sec^2 x \, dx \) into the integral:\[ \int e^{\tan x} \sec^2 x \, dx = \int e^u \, du \]

Step 4: Integrate

The integral of \( e^u \) is straightforward:\[ \int e^u \, du = e^u + C \] where \( C \) is the constant of integration.

Step 5: Back-substitute

Replace \( u \) with \( \tan x \) to revert to the original variable:\[ e^u + C = e^{\tan x} + C \] Thus, the indefinite integral is \( e^{\tan x} + C \).

Key Concepts

Understanding U-Substitution

U-substitution is a powerful technique used in integration to simplify complex expressions. It is particularly useful when you notice a function inside the integral that has its derivative also present in the integrand. This method is often likened to the reverse chain rule from differentiation.

The idea is to substitute a part of the integrand with a new variable, traditionally labeled as "u". This transformation:

• Makes the integral more straightforward and manageable by reducing it to a simpler form.
• Allows us to use standard integration formulas easily.

In the exercise given, you notice that \( \tan x \) is connected to its derivative \( \sec^2 x \) in the integrand. This hints that setting \( u = \tan x \) will simplify the process. By differentiating \( u \) with respect to \( x \), you find \( \frac{du}{dx} = \sec^2 x \), or simply \( du = \sec^2 x \, dx \). This substitution transforms the integral into \( \int e^u \, du \), making it straightforward to solve.

The Art of Integration

Integration is the process of finding an integral, which is the reverse operation of differentiation. It involves finding the function whose derivative is the given function. Indefinite integrals, like the one in this exercise, do not have specified limits and result in a general solution including a constant of integration denoted by \( C \).

Integrating functions can sometimes be challenging, especially when dealing with complex expressions. However, by using substitution methods such as u-substitution, the task can often be simplified significantly. After substituting, we end up with basic integrals that are easier to solve using known integration rules. In this case, the integral \( \int e^u \, du \) is well-known and solves to \( e^u + C \).

This step of integrating is where the transformation facilitates direct application of integration rules to solve the expression.

Trigonometric Functions in Integration

Trigonometric functions are often intertwined in calculus problems due to their ubiquitous applications and unique properties. In integration, recognizing these functions and their derivatives is crucial for applying techniques like u-substitution effectively.

Key trigonometric functions like \( \tan x \), \( \sec x \), \( \sin x \), and \( \cos x \) have derivatives that are commonly involved in integrals. For instance, the derivative of \( \tan x \) is \( \sec^2 x \), which was integral to simplifying our original problem using u-substitution.

Understanding the relationships between these functions and their derivatives enables quick transitions from complex to simple integrals. When you encounter expressions involving trigonometric functions, identifying these relationships helps unlock the solution path. In our exercise, this awareness allowed the transformation to \( \int e^u \, du \), leading to a simple integration solution.