Problem 43 - Thomas Calculus

Question

Evaluate each integral in Exercises \(43-46\) by using trigonometric identities and substitutions to reduce it to standard form. $$ \int(\sec x+\cot x)^{2} d x $$

Step-by-Step Solution

Answer

\(\int (\sec x + \cot x)^2 dx = \tan x - x - \cot x + 2 \ln |\sin x| + C\)

Step 1: Expand the Squared Expression

First, we need to expand the expression \((\sec x + \cot x)^2\). Use the binomial expansion formula \((a+b)^2 = a^2 + 2ab + b^2\).Substitute \(a = \sec x\) and \(b = \cot x\):\[(\sec x + \cot x)^2 = \sec^2 x + 2\sec x \cot x + \cot^2 x\]

Step 2: Simplify Using Trigonometric Identities

Replace the terms using trigonometric identities:- \(\sec^2 x = 1 + \tan^2 x\) (from the identity \(1 + \tan^2 x = \sec^2 x\))- \(\cot^2 x = \csc^2 x - 1\) (from the identity \(1 + \cot^2 x = \csc^2 x\))Substituting these:\[\sec^2 x = 1 + \tan^2 x\]\[\cot^2 x = \csc^2 x - 1\]

Step 3: Combine Terms

Combine the terms together:\[ \sec^2 x + \cot^2 x + 2\sec x \cot x = (1 + \tan^2 x) + (\csc^2 x - 1) + 2\sec x \cot x\] Simplifying gives:\[ \tan^2 x + \csc^2 x + 2\sec x \cot x\]

Step 4: Evaluate the Integral

Now evaluate the integral by splitting it into parts:\[\int \tan^2 x \, dx + \int \csc^2 x \, dx + \int 2\sec x \cot x \, dx\]- For \(\int \tan^2 x \, dx\), use the substitution \(\int \tan^2 x \, dx = \int (\sec^2 x - 1) \, dx = \tan x - x + C_1\).- For \(\int \csc^2 x \, dx\), the integral is directly \(-\cot x + C_2\).- For \(\int 2\sec x \cot x \, dx\), it becomes \(2 \ln |\sin x| + C_3\) (derivative of \(\sec x\) is \(\sec x \tan x\), hence \(\sec x \cot x = (\sin x)^{-1}\)).

Step 5: Combine Integral Results

Add the results from the separate integrals:\[(\tan x - x) + (-\cot x) + 2 \ln |\sin x| + C\] Simplifying, the integral evaluates to:\[\tan x - x - \cot x + 2 \ln |\sin x| + C\]where \(C\) is the constant of integration.

Key Concepts

Trigonometric Identities

Trigonometric identities help simplify integrals by transforming complex expressions into more manageable forms. These identities describe relationships between trigonometric functions. For example, the identity \( \sec^2 x = 1 + \tan^2 x \) derives from the Pythagorean theorem. Knowing this, you can exchange \( \sec^2 x \) for \( 1 + \tan^2 x \) in integrals, simplifying the calculation. Similarly, \( \cot^2 x \) is related to \( \csc^2 x \) through another identity: \( 1 + \cot^2 x = \csc^2 x \), allowing \( \cot^2 x \) to express as \( \csc^2 x - 1 \). Recognizing and applying these identities effectively is a pivotal skill in trigonometric integration, making the resulting calculations more straightforward and less time-consuming. Understanding these relationships opens paths to simplifying the integral into a standard form.

Trigonometric Substitutions

Trigonometric substitutions transform integrals into more solvable expressions using trigonometric functions. This technique often applies when expressions involve squared terms, particularly since many trigonometric identities deal with squares. For example, to evaluate an integral with \( \tan^2 x \), you could use \( \tan^2 x = \sec^2 x - 1 \) to rewrite and simplify the integral. These substitutions streamline the evaluations by aligning the integral with known derivatives of trigonometric functions. By converting original expressions into ones involving sine and cosine, it becomes easier to integrate using their known antiderivatives. Through trigonometric substitutions, you'll also develop insight into how different trigonometric functions interrelate, which can be beneficial in more complex integration problems.

Binomial Expansion

The binomial expansion formula is a vital algebraic tool in trigonometric integration, especially when dealing with expressions raised to powers such as \((a+b)^2 = a^2 + 2ab + b^2\). In the given problem, expanding \((\sec x + \cot x)^2\) helps break the expression into simpler parts that can be handled separately. By substituting \(a = \sec x\) and \(b = \cot x\), the expression becomes \( \sec^2 x + 2\sec x \cot x + \cot^2 x \). This transformation allows each term to be simplified or integrated individually. Utilizing binomial expansion makes intricate mathematical expressions more accessible, paving the way for easier and quicker solutions in both algebra and calculus contexts.

Integral Calculus

Integral calculus revolves around finding functions whose derivative is given, a process known as integration. When integrating trigonometric functions, the goal is to find the antiderivative, which represents the area under the curve of the original function. In the problem at hand, once simplified using identities, the evaluation splits into parts: \( \int \tan^2 x \, dx \), \( \int \csc^2 x \, dx \), and \( \int 2\sec x \cot x \, dx \). Each has a known antiderivative: \( \tan x - x + C_1 \) for \( \tan^2 x \), \( -\cot x + C_2 \) for \( \csc^2 x \), and \( 2 \ln |\sin x| + C_3 \) for \( 2\sec x \cot x \). These results combine into one comprehensive solution. Integral calculus, through parts like substitution, powers series, and trigonometric functions, allows for precise evaluations and provides a foundational component for understanding more complex mathematical models.