Problem 22 - Calculus Early Transcendentals

Question

Evaluate the following integrals. $$\int \frac{e^{\sin x}}{\sec x} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the following integral: $$\int \frac{e^{\sin x}}{\sec x} d x$$ Answer: The integral cannot be expressed in terms of elementary functions, but it can be represented using the Elliptic Integral of the Second Kind: $$\int \frac{e^{\sin x}}{\sec x} d x = E(\sin x, - \sqrt{-1}) + C$$

Step 1: Perform substitution

Let \(u = \sin x\). Therefore, the differential \(du\) is given by: $$du = \cos x dx$$ We also need to express \(\sec x\) in terms of \(u\). Recall that \(\sec x = \frac{1}{\cos x}\) and \(\sin^2 x + \cos^2 x = 1\). So, in terms of \(u\), we have: $$\cos^2 x = 1 - u^2 \Rightarrow \cos x = \sqrt{1 - u^2}$$ Hence, the integral becomes: $$\int \frac{e^{\sin x}}{\sec^h5x} d x = \int \frac{e^u}{\sqrt{1 - u^2}} du$$

Step 2: Integrate with respect to u

Now, we need to evaluate the integral: $$\int \frac{e^u}{\sqrt{1 - u^2}} du$$ Unfortunately, there is no elementary function representing the antiderivative of this integral. However, we can express this integral in terms of a special function called the "Elliptic Integral of the Second Kind." Using the definition of the Elliptic Integral of the Second Kind, the integral becomes: $$E(u, k) = \int \sqrt{1 - k^2 \sin^2 u} \, du$$ Where \(k\) is a constant. Comparing this definition with the given integral, we have: $$\int \frac{e^u}{\sqrt{1 - u^2}} du = \int \frac{e^u}{\sqrt{1 - u^2}} \cdot \frac{1}{\sqrt{1 - u^2}} du$$ $$= \int e^u \sqrt{1 - u^2} du = E(u, - \sqrt{-1})$$

Step 3: Substitute back the original variable

Now, we substitute back the original variable \(x\). Recall that \(u = \sin x\). So, the integral becomes: $$E(\sin x, - \sqrt{-1}) + C$$ Where \(C\) is the constant of integration. Thus, the final answer is: $$\int \frac{e^{\sin x}}{\sec x} d x = E(\sin x, - \sqrt{-1}) + C$$

Key Concepts

Substitution Method

The substitution method is a powerful technique in integral calculus used to simplify complex integrals. It works by introducing a new variable that replaces a part of the original function, making the integral easier to solve. For instance, in the integral \( \int \frac{e^{\sin x}}{\sec x} \, dx \), the substitution \( u = \sin x \) was introduced. This substitution helps in several ways:

• Changes variables to simplify the integrand.
• Makes the integral easier to approach, especially for complex expressions.
• Often converts trigonometric and exponential expressions into polynomial form.

By substituting \( u = \sin x \), we rewrote \( du = \cos x \, dx \), resulting in a simpler expression \( \frac{e^u}{\sqrt{1-u^2}} du \). This new integral form is easier to analyze and paves the way for solving the problem effectively.

Elliptic Integral

Elliptic integrals are special functions that cannot be expressed with standard elementary functions. They often arise in problems of arc length, areas under curves, and other geometrically challenging situations. The specific type of elliptic integral encountered in this exercise is related to the Elliptic Integral of the Second Kind. This can be defined as:\[ E(u, k) = \int \sqrt{1 - k^2 \sin^2 u} \, du \]The integral \( \int \frac{e^u}{\sqrt{1-u^2}} du \) is expressed in terms of this form using the parameter \( k = -\sqrt{-1} \). This resource helps to conclude the integral in a functionally appropriate way since there is no elementary solution. Understanding elliptic integrals is crucial in many advanced calculus problems where standard methods do not suffice.

Differential Calculus

Differential calculus is the study of rates at which quantities change. It's fundamental in understanding how to manipulate integrals, especially during substitution where differentials play a crucial role. In our scenario, recognizing the differential \( du = \cos x \, dx \) given \( u = \sin x \) is critical.Differential calculus concepts allow us:

• To change variables effectively, aligning the new integrand with the new limits or functionalities.
• To compute derivatives that help construct the differential \( du \).
• To understand changes in the variable, which leads to simplifying the integration process.

Thus, the mastery of differential calculus is essential for correctly implementing substitutions and solving the integral fully. It ensures that the substitutions are made smoothly, maintaining the integrity of the original function.