Question

Perform the indicated integrations. $$ \int \frac{\tan x}{\sqrt{\sec ^{2} x-4}} d x $$

Short answer

The integral simplifies to \( \sqrt{3} \sec^{-1}\left( \frac{\tan x}{\sqrt{3}} \right) + C \).

Step-by-step solution

Trigonometric Identity Substitution

Our task is to solve the integral \( \int \frac{\tan x}{\sqrt{\sec^2 x - 4}} \, dx \). First, recall the identity \( \sec^2 x = 1 + \tan^2 x \). Substitute this into the integral to transform it. The integral becomes \( \int \frac{\tan x}{\sqrt{1 + \tan^2 x - 4}} \, dx \). Simplify under the square root to get \( \int \frac{\tan x}{\sqrt{\tan^2 x - 3}} \, dx \).

Substitution Method

Set \( u = \tan x \), then \( du = \sec^2 x \, dx \). From \( \sec^2 x = 1 + \tan^2 x = 1 + u^2 \), we have \( dx = \frac{du}{1+u^2} \). Substitute \( u \) and \( dx \) into the integral: \( \int \frac{u}{\sqrt{u^2 - 3}} \cdot \frac{du}{1+u^2} \).

Simplify the Integral

The integral simplifies to \( \int \frac{u}{(1+u^2)\sqrt{u^2 - 3}} \, du \). This can be rewritten as \( \int \frac{1}{\sqrt{u^2 - 3}} \, du - \int \frac{u^2}{(1+u^2)\sqrt{u^2 - 3}} \, du \). The first part involves the integral \( \int \frac{1}{\sqrt{u^2 - 3}} \, du \), and suggests a hyperbolic substitution.

Hyperbolic Substitution

Let \( u = \sqrt{3} \sec(\theta) \). Then \( du = \sqrt{3} \sec(\theta)\tan(\theta) \, d\theta \) and \( \sqrt{u^2 - 3} = \sqrt{3} \tan(\theta) \). The integral becomes \( \int \frac{\sqrt{3} \sec(\theta)}{\sqrt{3} \tan(\theta)} \cdot \sqrt{3} \sec(\theta)\tan(\theta) \, d\theta \). The expression simplifies nicely.

Final Integration

After simplification, the remaining integral is \( \sqrt{3} \int d\theta \), which evaluates to \( \sqrt{3} \theta + C \), where \( C \) is the constant of integration. To convert back to \( u \), remember that \( u = \sqrt{3} \sec(\theta) \), so \( \theta = \sec^{-1}(\frac{u}{\sqrt{3}}) \). Substitute \( u = \tan x \) to find \( \theta = \sec^{-1}(\frac{\tan x}{\sqrt{3}}) \).

Simplifying Results and Final Answer

The full solution in terms of \( x \) is \( \sqrt{3} \sec^{-1}\left( \frac{\tan x}{\sqrt{3}} \right) + C \). This is the final answer after performing all substitutions and simplifications.

Key concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in calculus and particularly useful in trigonometric integration. They help simplify complex expressions into more workable forms. In the solution provided, the key identity used is \( \sec^2 x = 1 + \tan^2 x \). This reveals the underlying relationships between trigonometric functions and aids in transforming the original integral.
By applying this identity, we restructure the expression \( \sqrt{\sec^2 x - 4} \) into \( \sqrt{1 + \tan^2 x - 4} \), simplifying it to \( \sqrt{\tan^2 x - 3} \). Understanding and manipulating these identities allow trigonometric functions to be expressed in terms of one another, streamlining the integration process.

Substitution Method

The substitution method in integration is a powerful technique designed to simplify the integrand by changing variables. This transforms a complex integral into a simpler one that is easier to evaluate. In our example, we set \( u = \tan x \), which implies \( du = \sec^2 x \, dx \). By substitution, the integral transforms into an easily manageable form.
This step involves replacing the variable \( x \) with \( u \), and expressing \( dx \) in terms of \( du \) using the relation \( dx = \frac{du}{1+u^2} \). This method not only helps in breaking down the integral but also paves the way for applying further techniques like hyperbolic substitution.

Hyperbolic Substitution

Hyperbolic substitution is useful for integrating expressions involving square roots, especially when they contain terms like \( u^2 - a^2 \). This substitution leverages hyperbolic functions which have properties similar to trigonometric functions. In this scenario, we use \( u = \sqrt{3} \sec(\theta) \), which implies that \( du = \sqrt{3} \sec(\theta)\tan(\theta) \, d\theta \).
A key observation here is that \( \sqrt{u^2 - 3} = \sqrt{3} \tan(\theta) \). By applying this substitution, the integral simplifies drastically, allowing it to eventually be expressed in terms of \( \theta \), leading to more straightforward integration. This substitution often results in canceling terms and simplifying expressions, making the integration path more direct.

Integration Techniques

Integration techniques encompass methods and strategies used to solve integrals effectively. While fundamental techniques like substitution help simplify integrals, in this solution, it's evident that breaking down the problem into manageable parts is crucial. The original integral splits into two parts: \( \int \frac{1}{\sqrt{u^2 - 3}} \, du \) and \( \int \frac{u^2}{(1+u^2)\sqrt{u^2 - 3}} \, du \).
Approaching each term separately enables us to apply different strategies like the hyperbolic substitution and to identify patterns and symmetries. By reducing the complexity of the integral into simpler forms, we ensure the solution process becomes systematic and less prone to error. Mastering these techniques not only aids in solving integrals efficiently but also enhances one's analytical skills in handling complex mathematical expressions.