Problem 60 - Calculus Single Variable

Question

The denominator of the given integrand is of the form \(a \pm b\). Multiply numerator and denominator by \(a \mp b\) to obtain a difference of squares in the denominator. Then use an appropriate trigonometric identity before integrating. $$ \int \frac{1}{\sec (x)-1} d x $$

Step-by-Step Solution

Answer

Transform using the conjugate, integrate, final integral challenges the direct use in elementary functions.

Step 1: Identify the Form

The integrand is \( \frac{1}{\sec(x) - 1} \). Notice that the denominator is of the form \( a - b \) where \( a = \sec(x) \) and \( b = 1 \).

Step 2: Multiply by Conjugate

To rationalize the denominator and form a difference of squares, multiply both numerator and denominator by the conjugate of the denominator: \( \sec(x) + 1 \). This transforms the integral into:\[ \int \frac{\sec(x) + 1}{(\sec(x) - 1)(\sec(x) + 1)} \, dx \]

Step 3: Simplify Denominator Using Difference of Squares

The expression \((\sec(x) - 1)(\sec(x) + 1) \) simplifies to \( \sec^2(x) - 1^2 \) or \( \sec^2(x) - 1 \), which equals \( \tan^2(x) \). Thus the integral becomes:\[ \int \frac{\sec(x) + 1}{\tan^2(x)} \, dx \]

Step 4: Split the Integral

Separate the integrand into two parts:\[ \int \frac{\sec(x)}{\tan^2(x)} \, dx + \int \frac{1}{\tan^2(x)} \, dx \]

Step 5: Use Trigonometric Identities

The identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) gives \( \tan^2(x) = \frac{\sin^2(x)}{\cos^2(x)} \). Therefore, \( \frac{1}{\tan^2(x)} = \frac{\cos^2(x)}{\sin^2(x)} \) which is \( \cot^2(x) \). Also recognize \( \frac{\sec(x)}{\tan^2(x)} = \frac{1}{\sin(x) \cos(x)} = \csc(x) \sec(x) \).

Step 6: Simplify Each Integral

Simplify:- \( \int \csc(x) \sec(x) \, dx \)- \( \int \cot^2(x) \, dx \)Using the identities and known integral results.

Step 7: Solve Simplified Integrals

The integral of \( \csc(x) \sec(x) \) does not resolve into standard elementary functions easily, and may require substitution for simplification. Similarly, the integral of \( \cot^2(x) \) can be rewritten and solved as \( -\int csc^2(x) dx + \int dx \), yielding:\[ -\cot(x) + x + C \]

Step 8: Write the Final Solution

Combine the results, but note simplifications may lead to expressions typically integrated by particular methods or Wil be provided under assumed conditions.

Key Concepts

Integration Techniques

Integration is a fundamental concept in calculus used to find areas under curves, solve differential equations, and compute total accumulated quantities. In this exercise, we explore specific techniques to solve integrals involving trigonometric expressions. Integration techniques often involve simplifying the integrand through various methods, such as substitution, partial fraction decomposition, or using algebraic identities.

In our original exercise, one important technique is the multiplication by the conjugate to simplify the integrand. This technique is common when dealing with expressions that can form a difference of squares, allowing for easier integration by breaking down complex fractions. Another common technique is the separation of integrals, where a complex integral is split into two or more simpler integrals that are easier to evaluate. This often involves algebraic manipulation and recognizing common mathematical patterns or identities.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. They are useful for simplifying expressions and solving integrals or equations that contain trigonometric terms. Some key identities used in integration include:

• Pythagorean identities: - \( an^2(x) + 1 = ext{sec}^2(x) \) - \( ext{sec}^2(x) - ext{tan}^2(x) = 1 \)
• Reciprocal identities: - \( ext{sec}(x) = \frac{1}{ ext{cos}(x)} \)
• Quotient identities: - \( ext{tan}(x) = \frac{ ext{sin}(x)}{ ext{cos}(x)} \)

In this particular exercise, the identity \( an^2(x) + 1 = ext{sec}^2(x) \) is used. This identity allows the transformation of the denominator from a complex trigonometric expression into a simpler form, facilitating integration.

Furthermore, knowing these identities can help in rewriting integrals in ways that are more easily evaluated or match standard integral forms, enhancing your problem-solving strategies in integral calculus.

Difference of Squares

The concept of difference of squares is a powerful algebraic tool used to simplify complex expressions. It states that for any two terms \(a\) and \(b\), the expression \(a^2 - b^2\) can be factored as \( (a - b)(a + b) \).

In our problem, the denominator \( ext{sec}(x) - 1 \) is multiplied by \( ext{sec}(x) + 1 \) to form a difference of squares. This multiplication gives us a new denominator of \( ext{sec}^2(x) - 1 \), which simplifies to \( an^2(x) \). This transformation simplifies the integrand drastically and allows for straightforward integration because the expression \( an^2(x) \) is easier to work with than its original form.

Understanding how to identify and use the difference of squares helps in recognizing when and how you can simplify expressions to make them more manageable, especially in integral calculus where simplifying the integrand is a key step.

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify the integration of expressions involving radical terms or specific trigonometric functions by substituting a suitable trigonometric identity or expression. The substitution typically reduces the problem to an integral involving simpler trigonometric functions.

In this exercise, trigonometric substitution is necessary when manipulating trigonometric functions such as \( ext{sec}(x) \), \( an(x) \), and their combinations. By using identities like \( an^2(x) = rac{ ext{sin}^2(x)}{ ext{cos}^2(x)} \), we can transform the integrand into expressions like \( rac{1}{ ext{sin}(x) ext{cos}(x)} \), allowing for easier integration using known substitution techniques or identities.

This approach is particularly useful when dealing with integrals that do not have straightforward antiderivatives or expressions that can easily be integrated in their original form. Hence, mastering trigonometric substitution can significantly enhance your toolkit for tackling complex integral problems in calculus.