Problem 70 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(47-70\) $$ \int \frac{\sec ^{2} y d y}{\sqrt{1-\tan ^{2} y}} $$

Step-by-Step Solution

Answer

\( \int \frac{\sec ^{2} y \, dy}{\sqrt{1-\tan ^{2} y}} = \sin^{-1}(\tan y) + C \).

Step 1: Understand the Integral

We are given \[\int \frac{\sec ^{2} y \, d y}{\sqrt{1-\tan^{2} y}}\]Our goal is to evaluate this integral. It is beneficial to recognize how the expressions involving trigonometric identities can simplify the integral.

Step 2: Use Substitution

Notice that \( \tan^2 y + 1 = \sec^2 y \). This is the standard Pythagorean trigonometric identity. With \( 1 - \tan^2 y = \sec^2 y - 1 \), the integral simplifies using this substitution:Let \( u = \tan y \implies du = \sec^2 y \, dy \).The integral becomes \[\int \frac{du}{\sqrt{1-u^2}}\]

Step 3: Simplify the Integral

With the substitution complete, our integral is now \[\int \frac{du}{\sqrt{1-u^2}}\]This is a standard integral known as the arcsine integral, which evaluates to \[ \sin^{-1}(u) + C \].

Step 4: Substitute Back

Convert back to the original variable, \( y \), using \( u = \tan y \):\[\sin^{-1} (\tan y) + C\]

Step 5: Final Expression

Thus, the evaluated integral is:\[\sin^{-1}(\tan y) + C\] where \( C \) is the constant of integration.

Key Concepts

Trigonometric Identities

When dealing with integrals involving trigonometric functions, trigonometric identities play a crucial role in simplification. These identities help rewrite complex expressions into forms that are easier to integrate. In our example, we utilized a key trigonometric identity:

• The Pythagorean identity, which is \( \tan^2 y + 1 = \sec^2 y \).

This identity is fundamental and is derived from the definition of tangent and secant in terms of sine and cosine. Recognizing and applying these identities can transform an intricate integral into a more manageable one. Keeping these identities in mind while solving trigonometric integrals is vital.

Integration Techniques

Integration is a fundamental tool in calculus, and mastering different techniques can greatly simplify the problem-solving process. One of the most useful techniques is substitution, which is often employed when dealing with trigonometric integrals. Here, we initially focused on recognizing the structure of the integral and how it relates to known trigonometric identities. After identifying the identity \( 1 - \tan^2 y = \sec^2 y - 1 \), it became clear that a substitution involving \( \tan y \) would simplify the integral. This insight significantly reduces the complexity and transforms it into a more recognizable form, such as an arcsine integral. Always aim to identify opportunities where substitution can streamline the integration process.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry, relating the squares of sine, cosine, and tangent to one another. In the realm of integration, it allows for the transformation of expressions that seem insurmountable at first glance.The three fundamental forms of the Pythagorean identity are:

• \( \sin^2 y + \cos^2 y = 1 \)
• \( \tan^2 y + 1 = \sec^2 y \)
• \( 1 + \cot^2 y = \csc^2 y \)

In our exercise, we used the form \( \tan^2 y + 1 = \sec^2 y \) to aid in substitution. Recognizing which form of the identity to apply can profoundly simplify the integration process, changing the problem from one that is complex to one that is well-understood.

Substitution Method

The substitution method, often referred to as \( u \)-substitution, is a powerful technique for solving integrals. It focuses on transforming the integral into a more straightforward form, typically one that is recognizable as a standard integral.The substitution method involves:

• Selecting a substitution that simplifies part of the integral expression.
• Changing the variable of integration from \( y \) to \( u \) with the relation \( u = \tan y \), where \( du = \sec^2 y \, dy \).
• Rewriting the integral in terms of \( u \), thus transforming it into the simpler form \( \int \frac{du}{\sqrt{1-u^2}} \), which is a known integral that evaluates to \( \sin^{-1}(u) \).

After applying the substitution method, remember to convert back to the original variable to express the final solution correctly, which in this case results in \( \sin^{-1}(\tan y) + C \). Being comfortable with choosing appropriate substitutions can substantially reduce the effort required to solve integrals, leading to a path of least resistance towards the correct answer.