Problem 13 - Calculus: Early Transcendental Functions

Question

Find the general antiderivative. $$\int 2 \sec x \tan x \, d x$$

Step-by-Step Solution

Answer

The antiderivative of \(\int 2 \sec x \tan x \ dx\) is \(2 \sec x + C\).

Step 1: Recognize the link between the secant and tangent functions

The derivatives of secant function, \(\sec x\), is \(\sec x \tan x\). In other words, \(d/dx [ \sec x ] = \sec x \tan x\). Our integral function exactly matches this pattern, but scaled by a factor of 2.

Step 2: Use the chain rule of calculus

According to the chain rule, if we have a function \(y = f(g(x))\), then the derivative of \(y\) with respect to \(x\) is given by \(dy/dx = f'(g(x)) * g'(x)\). The chain rule can be applied to the integral in reverse to find the antiderivative. Since \(d/dx [ \sec x ] = \sec x \tan x\), we can apply the chain rule in reverse to find that the antiderivative of the integral is a multiple of \(\sec x\).

Step 3: Apply the integral rule

We obtained that the antiderivative of \(\sec x \tan x\) is \(\sec x\). Therefore, the antiderivative of our function \(\int 2 \sec x \tan x \ dx\) is equal to \(2 \int \sec x \tan x \ dx = 2 \sec x\), plus the integration constant \(C\).

Key Concepts

Secant Function

In trigonometry, the secant function is the reciprocal of the cosine function. The cosine gives the ratio of the adjacent side over the hypotenuse in a right triangle. Thus,

• The secant function, represented as \( \sec x \), gives the reciprocal, making it hypotenuse over adjacent.
• This means \( \sec x = \frac{1}{\cos x} \).

Understanding this relationship is crucial when working with integrals and derivatives involving secant and its related function, tangent. In calculus, derivatives focus on the rate of change. The derivative of secant, \( \sec x \), results in \( \sec x \tan x \). Whenever you see \( \sec x \tan x \), recall it is the derivative of the secant. This allows you to reverse the process in integration, identifying that when you integrate \( \sec x \tan x \), the result should logically bring you back to \( \sec x \).
Thus, recognizing function relationships like these clears up the process of finding antiderivatives.

Tangent Function

The tangent function is another basic trigonometric function, often introduced alongside sine and cosine. It is defined based on a right triangle, specifically as the ratio of the opposite side to the adjacent side.

• The formula is \( \tan x = \frac{\sin x}{\cos x} \).
• This highlights how tangent is dependent on sine and cosine.

A critical aspect of the tangent is its relationship with the secant in calculus. When performing differentiation, the product \( \sec x \tan x \) emerges as the derivative of \( \sec x \).
This derivative relationship serves as a pivotal tool in solving integrals, where recognizing \( \tan x \) paired with \( \sec x \) directly brings insights into antiderivatives. The familiarity with how these functions work together aids in efficiently solving calculus problems that involve these trigonometric expressions.

Chain Rule

In calculus, the chain rule is a fundamental principle used to find derivatives of composite functions. When faced with a function comprising one function inside another, the chain rule helps break it down.

• Say you have \( y = f(g(x)) \), the derivative is \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).
• The chain rule helps to manage this by differentiating the outer function and the inner function separately.

In the context of integration, the chain rule can be reversed to solve for an antiderivative. This means identifying an inner function whose derivative is present in the integrand, thus seeing how it fits in a larger picture of differentiating and integrating functions. Using the chain rule in reverse, you can determine that \( \int 2 \sec x \tan x \, dx \) simplifies due to recognizing \( \sec x \tan x \) as the derivative of \( \sec x \). This understanding provides a straightforward path to calculating the antiderivative accurately.