Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

Perform the indicated integrations. $$ \int \frac{\sec ^{3} x+e^{\sin x}}{\sec x} d x $$

Step-by-Step Solution

Answer

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

Step 1: Simplify the integrand

The given integrand is \( \frac{\sec ^{3} x + e^{\sin x}}{\sec x} \). We can separate this expression into two parts: \( \frac{\sec ^{3} x}{\sec x} \) and \( \frac{e^{\sin x}}{\sec x} \). This simplifies to \( \sec^2 x \) and \( e^{\sin x} \cos x \), respectively. Therefore, the integral becomes \( \int ( \sec^2 x + e^{\sin x} \cos x) \, dx \).

Step 2: Integrate each term separately

We'll integrate the two resulting terms separately: 1. \( \int \sec^2 x \, dx \) The integral of \( \sec^2 x \) with respect to \( x \) is \( \tan x \) because the derivative of \( \tan x \) is \( \sec^2 x \).2. \( \int e^{\sin x} \cos x \, dx \) To solve this, use substitution by letting \( u = \sin x \), which implies \( du = \cos x \, dx \). Therefore, the integral becomes \( \int e^u \, du \), which integrates to \( e^u \). Substituting back \( u = \sin x \), we get \( e^{\sin x} \).

Step 3: Combine the results

Combine the two integrated parts from Step 2. The integral of the whole expression is:\[ \tan x + e^{\sin x} + C \]where \( C \) is the constant of integration, accounting for an indefinite integral.

Key Concepts

Integration Techniques

Integration techniques are essential tools in calculus to solve integrals that cannot be directly computed. One common technique is breaking down the integrand, which becomes particularly useful when an integrand is complex or can be separated into simpler parts.
In the provided exercise, you see this technique in action where the integrand \( \frac{\sec ^{3} x+e^{\sin x}}{\sec x} \) is split into two simpler expressions: \( \sec^2 x \) and \( e^{\sin x} \cos x \). This is essentially a form of algebraic manipulation.

• The first part, \( \sec^2 x \), is already in a form that is easier to integrate.
• The second part, \( e^{\sin x} \cos x \), suggests a substitution method might be effective.

These strategies simplify the integral and help in approaching each term individually.

Trigonometric Integrals

Trigonometric integrals are those involving trigonometric functions such as sine, cosine, tangent, and their inverses. In this exercise, the term \( \sec^2 x \) represents a trigonometric integral.
The integration of \( \sec^2 x \) is straightforward because it is a standard result in integral calculus. Since we know that the derivative of \( \tan x \) is \( \sec^2 x \), the integral of \( \sec^2 x \) is \( \tan x \).
Understanding these standard results in calculus can drastically simplify problems with trigonometric integrals:

• Knowing that \( \int \sec^2 x \, dx = \tan x + C \), where \( C \) is the constant of integration.
• It helps in quickly identifying the result without elaborate steps.

These standard trigonometric integrals are foundational for solving more complex problems.

Substitution Method

The substitution method is a powerful tool for solving integrals, especially when the integral contains a composite function. Essentially, this technique changes the variable into one that simplifies the integral.
In our exercise, we deal with the part \( \int e^{\sin x} \cos x \, dx \). Here, \( e^{\sin x} \) is a composite function, indicating that substitution could simplify integration. The substitution \( u = \sin x \) leads to \( du = \cos x \, dx \) and transforms the integral to \( \int e^u \, du \).

• This substitution helps simplify complex integrands by reducing them to basic forms.
• We then integrate \( \int e^u \, du \), which is known to be \( e^u + C \).

Finally, substitute back \( u = \sin x \) to obtain the result \( e^{\sin x} + C \). This method is invaluable for working through different integral types.

Indefinite Integrals

Indefinite integrals represent a class of integrals without specified limits, aiming to find an antiderivative, a function that when differentiated gives the original function.
The result of an indefinite integral is typically expressed with a \( + C \), where \( C \) is the constant of integration. This accounts for all possible vertical shifts in the antiderivative function.
In the solved exercise, the integral \( \int ( \sec^2 x + e^{\sin x} \cos x) \, dx \) is an indefinite one. The result, \( \tan x + e^{\sin x} + C \), shows combined antiderivatives of individual components:

• The \( \tan x \) comes from integrating \( \sec^2 x \).
• The \( e^{\sin x} \) follows from the substitution and integration of \( e^{\sin x} \cos x \).

All indefinite integrals include this constant \( C \) since differentiation of a constant yields zero, demonstrating the nature of general solutions in calculus.