Problem 32 - Essential Calculus

Question

Find the integral involving secant and tangent. $$ \int \frac{\tan ^{2} x}{\sec ^{5} x} d x $$

Step-by-Step Solution

Answer

The solution to the integral is \( -\frac{2\tan{x}}{3(1 + \tan^{2}x)^{3/2}} + C \)

Step 1: Conversion of secants into tangents

Recognizing that \( \sec^2x = 1 + \tan^2x \), the given integral can also be written as \( \frac{\tan^{2}x}{(1 + \tan^{2}x)^{5/2}}dx \) by substituting this identity.

Step 2: Substitution

Make a substitution where \( u = \tan{x} \). Then, \( du = \sec^2x dx \). Now the integral equation becomes \( \int \frac{u^{2}}{(1 + u^{2})^{5/2}} du \).

Step 3: Solve the Integral

This integral can now be solved using the standard method by realizing that the numerator is the derivative of the denominator. The integral becomes \( -\frac{2u}{3(1 + u^{2})^{3/2}} + C \).

Step 4: Re-substitution

Substituting \( u = \tan{x} \) back into the integral equation gives: \( -\frac{2\tan{x}}{3(1 + \tan^{2}x)^{3/2}} + C \)

Key Concepts

Definite and Indefinite Integrals

Integration is a fundamental concept in calculus that comes in two main forms: definite and indefinite. An indefinite integral, often referred to as an antiderivative, represents a family of functions and includes a constant of integration, denoted as \( C \). For example, when we calculate the indefinite integral \( \int f(x) \, dx \), the result includes \( C \), reflecting the infinite number of possible antiderivatives.
In contrast, a definite integral is about finding the area under a curve between specific limits, say from \( a \) to \( b \). It does not involve \( C \) because it provides a specific numerical value representing this area. We denote it as \( \int_a^b f(x) \, dx \).
In the context of the original exercise, it focused on an indefinite integral as it seeks a general form of integration without limits. This procedure involves transforming the given expression into a more manageable form and integrating it using standard calculus techniques.

Trigonometric Substitution

Trigonometric substitution is a powerful technique used in integral calculus to tackle integrals involving expressions with squares, like \( \tan^2(x) \) or \( \sec^2(x) \). The substitution makes use of trigonometric identities to simplify the integral into a form that is easier to integrate.
In the step-by-step solution provided, the identity transformation \( \sec^2x = 1 + \tan^2x \) was used to rewrite the original integral. This approach helps simplify the terms by expressing them in terms of tangent, which is manageable due to its standard derivative properties.

• Step 1: Recognize that transforming one type of trigonometric function into another can simplify the expression significantly.
• Step 2: Perform a substitution, like \( u = \tan{x} \), to transform the integral into another variable, making it easier to handle.

This strategic substitution not only simplified the expression but also set the stage for applying further calculus methods to solve the integral.

Integral Calculus

Integral calculus is the branch of mathematics that focuses on the concept of integration. Understanding integrals is key to solving problems related to areas, volumes, and sums. In this process, integrals are used either to add parts together to find a whole (as with definite integrals) or to find a general formula for such summing operations (as with indefinite integrals).
In our exercise, we explored an indefinite integral involving trigonometric functions. The primary goal is to determine the antiderivative that fits the original expression. Key processes in integral calculus include recognizing derivative and integral relationships and the skillful use of substitution methods like the trigonometric substitution used in the step-by-step solution. Solving the integral means understanding the interaction between different calculus principles and applying them to break down complex expressions into solvable parts.
Integral calculus's principles enable us to understand how changes in one part of a mathematical expression can affect the whole system, a critical concept for fields such as physics, engineering, and beyond.