Problem 34 - Calculus of a Single Variable

Question

Finding an Indefinite Integral of a Trigonometric Function In Exercises \(31-40\) , find the indefinite integral. $$ \int \sec \frac{x}{2} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \( \sec \frac{x}{2} \) is \( 2 \ln |\sec \frac{x}{2} + \tan \frac{x}{2}| + C \)

Step 1: Multiplying the integrand by \(\sec \frac{x}{2} + \tan \frac{x}{2}\)

First, multiply and divide the integrand by \(\sec \frac{x}{2} + \tan \frac{x}{2}\): \[ \int \sec \frac{x}{2} dx = \int \frac{\sec \frac{x}{2} \cdot (\sec \frac{x}{2} + \tan \frac{x}{2})}{\sec \frac{x}{2} + \tan \frac{x}{2}} dx\]

Step 2: Using trigonometric identities

Apply the trigonometric identity \(\sec^2 \theta = 1 + \tan^2 \theta\): \[= \int \frac{\tan \frac{x}{2} + \sec^2 \frac{x}{2}}{\sec \frac{x}{2} + \tan \frac{x}{2}} dx\]

Step 3: Substitution

Let \(u = \sec \frac{x}{2} + \tan \frac{x}{2}\), then du = \(\frac{1}{2} \cdot (\sec \frac{x}{2} + \tan \frac{x}{2})\) dx, therefore our integral becomes: \[= 2 \int \frac{du}{u}\]

Step 4: Final Step

This is a basic form of integral and its integration gives: \[= 2\ln |u| + C\] Finally, substitute \(u = \sec \frac{x}{2} + \tan \frac{x}{2}\) back into the equation: \[= 2 \ln |\sec \frac{x}{2} + \tan \frac{x}{2}| + C\]

Key Concepts

Integration Techniques

Integration is a fundamental technique in calculus used to find the antiderivative or the area under a curve of a function. There are several methods of integration, but some of the most widely used include the power rule, integration by parts, trigonometric integration, partial fraction decomposition, and the substitution method. Utilizing these techniques often requires some preliminary steps such as algebraic manipulation or trigonometric identities to simplify the integrand.

For the case of integrating trigonometric functions, like the given exercise \( \int \sec \frac{x}{2} dx \), pairing it with trigonometric identities or other forms can help to achieve a format that’s easier to integrate. The exercise incorporates a clever multiplication by \( \sec \frac{x}{2} + \tan \frac{x}{2} \), which doesn’t change the function’s value but enables the use of known identities that simplify the integral.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables where both sides of the equality are defined. These identities are immensely useful in simplifying trigonometric expressions and solving trigonometric equations. Key identities include the Pythagorean identities, the angle sum and difference formulas, double and half-angle formulas, and the identities for the reciprocal trigonometric functions.

One of the most used identities is \( \sec^2 \theta = 1 + \tan^2 \theta \), which is a derived form of the Pythagorean identity. In the presented exercise, using this specific identity enables us to express the entire integrand in terms of 'u', which greatly simplifies the problem. This highlights the importance of knowing these identities and being able to recognize when and how to apply them for solving integration problems involving trigonometric functions.

Substitution Method

The substitution method, also known as u-substitution, is one of the primary techniques for finding the indefinite integral of a function. It's akin to the reverse of the chain rule in differentiation. The essence of the method is to simplify an integral by substituting a part of the integrand with a new variable 'u', which converts the original integral into an easier form that is typically straightforward to integrate.

In our example, setting \( u = \sec \frac{x}{2} + \tan \frac{x}{2} \) and finding \( du \) in terms of \( dx \) allows us to rewrite the integral \( \int \sec \frac{x}{2} dx \) in terms of 'u', simplifying it to a natural logarithmic form. Remember that after finding the integral in terms of 'u', it's essential to convert it back to the original variable, in this case by substituting 'u' back with \( \sec \frac{x}{2} + \tan \frac{x}{2} \). This method profoundly eases the integration process, particularly when dealing with composite functions and complex trigonometric integrands.