Problem 22 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Compute the following integrals. $$ \int \sec ^{2} x \tan ^{2} x d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{1}{2} * \tan^2 x + C \).

Step 1: Identify the function and its derivative

The function to be considered is \( \tan^2 x \) and its derivative is \( 2 \tan x \sec^2 x \)

Step 2: Rewrite the integral

Notice that the integral can be rewritten in terms of the identified function and its derivative. Therefore, it can be rewritten as \[ \frac{1}{2} \int 2 \tan x \sec^2 x \cdot \tan x d x\].

Step 3: Make a substitution

Let \( u = \tan x \). Then the derivative \( du = \sec^2 x dx \). Substitute this into the integral to get: \[ \frac{1}{2} \int 2 u du\]

Step 4: Solve the integral

Solving \( \frac{1}{2} \int 2 u du \), we get \( \frac{1}{2} * (u^2) \).

Step 5: Substitute back the original variable

Now substitute back the original variable \( u= \tan x \) into the result: \( \frac{1}{2} * \tan^2 x \)

Step 6: Add the constant of integration

The last step is adding the constant of integration, \( C \). Hence, the final result is: \( \frac{1}{2} * \tan^2 x + C \).

Key Concepts

Indefinite Integral

Indefinite integrals, often referred to simply as integrals, are a fundamental concept in calculus. They represent the reverse process of differentiation, meaning that taking the integral of a function's derivative will yield the original function (apart from a constant of integration). The general form of an indefinite integral is expressed as \( \int f(x) dx \) and the result is a family of functions with an added constant denoted by \( C \) because the process of differentiation wipes out any constant term. This constant represents all the possible vertical translations of the antiderivative.
While solving an indefinite integral, we aren’t given definite bounds over which to integrate, hence, we're looking for a general form of the antiderivative. The solution to the integral in our exercise, \( \frac{1}{2} * \tan^2 x + C \), is a prime example of an indefinite integral, showing the antiderivative of the given function with an added constant of integration, \( C \).

Integration by Substitution

Integration by substitution is a technique to simplify the integration process when the integrand (the function to be integrated) involves a composition of functions. It works by replacing a function within an integral with a new variable, thus transforming the integral into a simpler form that is easier to solve.

• Firstly, we choose a substitution variable, usually denoted as \( u \), which is a function of \( x \).
• Secondly, we calculate the differential \( du \), which represents the derivative of our substitution function multiplied by \( dx \).
• Finally, we replace all occurrences of the original \( x \) variable and \( dx \) in our integral with \( u \) and \( du \) accordingly, solving the simpler integral before substituting back to the original variable.

Our exercise showcased this method by substituting \( u = \tan x \) which simplified the integral \( \int \sec^2 x \tan^2 x dx \) to a much simpler form \( \int u^2 du \) that could then be integrated directly.

Secant Function

The secant function, denoted as \( \sec(x) \), is one of the trigonometric functions and is essentially the reciprocal of the cosine function, defined as \( \sec(x) = \frac{1}{\cos(x)} \). The secant function grows without bound as it approaches \( \frac{\pi}{2} + k\pi \), where \( k \) is an integer, since the cosine function approaches zero at these points.

In calculus, \( \sec^2(x) \) frequently appears as the derivative of the tangent function, \( \tan(x) \). This relationship is particularly useful in integration and differentiation problems involving trigonometric identities.

Tangent Function

The tangent function, \( \tan(x) \), is another fundamental trigonometric function, which can be represented as the ratio of the sine and cosine functions, \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). Its graph is cyclic with periods of \( \pi \) and has vertical asymptotes where the cosine function is zero.
Due to the quotient identity, \( \tan^2(x) \) plus 1 equals \( \sec^2(x) \) (\( \tan^2(x) + 1 = \sec^2(x) \)), which is a key trigonometric identity used in calculus. The derivative of \( \tan(x) \) is \( \sec^2(x) \), and this relationship is invaluable when evaluating integrals involving tangent functions, as seen in our exercise. Integrals with tangent functions often require methods like substitution to simplify them into a more easily integrable form.