Problem 4 - Technical Mathematics with Calculus

Question

Integrate $$\int \sec 2 \theta d \theta$$

Step-by-Step Solution

Answer

The integral \( \int \sec 2\theta \, d\theta \) is equal to \( \frac{1}{2}\ln|\sec 2\theta + \tan 2\theta| + C \).

Step 1: Recognize the integral form

Identify that the integral is of the form \( \int \sec x \, dx \), where \( x = 2\theta \). This integral cannot be solved in its current form using elementary functions. The typical approach is to multiply by a form of 1 that will make the expression integrable, using the identity \( \sec x = \frac{1}{\cos x} \).

Step 2: Multiply by a helpful form of 1

Multiply the secant function by \( \frac{\sec x + \tan x}{\sec x + \tan x} \), which is a form of 1. This technique is useful because it will simplify the integral into a form that we can solve. So, the integral becomes \( \int \sec^2 x (\sec x + \tan x) / (\sec x + \tan x) \, dx \).

Step 3: Simplify the expression

After multiplying, the integral simplifies to \( \int (\sec^2 x / (\sec x + \tan x)) (\sec x + \tan x) \, dx \). Now we can substitute \( u = \sec x + \tan x \), which makes \( du = (\sec x \tan x + \sec^2 x) \, dx \). The integrand now matches part of the \( du \) expression.

Step 4: Apply the substitution

Replace the integrand with the new variable to get \( \int 1/u \, du \). This is a straightforward integral that corresponds to the natural logarithm.

Step 5: Integrate using the natural logarithm

Integrating \( 1/u \) with respect to \( u \) gives \( \ln |u| + C \), where \( C \) is the constant of integration.

Step 6: Resubstitute the original variable

Resubstitute \( u \) with \( \sec x + \tan x \) to get the final integral in terms of the original variable.

Key Concepts

Trigonometric Integration

Trigonometric integration plays a vital role in calculus, especially when dealing with functions that are products of trigonometric and algebraic expressions. The integration of secant functions, in particular, is a common problem that students must learn to tackle. The secant function, \( \sec x = \frac{1}{\cos x} \), itself is not straightforward to integrate. Techniques such as multiplying by a clever form of one or using trigonometric identities are often employed to simplify the integral into a more manageable form.

In the context of our exercise, by multiplying \( \sec 2\theta \) by \( \frac{\sec 2\theta + \tan 2\theta}{\sec 2\theta + \tan 2\theta} \) we cleverly create a scenario where the numerator complements the derivative of the denominator, setting the stage for an easier integration via substitution. Understanding these manipulations is crucial not only for solving this particular integral but for a whole class of trigonometric integrals encountered in calculus.

Substitution Method in Integration

The substitution method, also known as \( u \) substitution, is a powerful tool in calculus for tackling complex integrals. When encountering an integral that is difficult to solve directly, identifying a part of the integrand that can be substituted with a new variable \( u \) can simplify the expression significantly.

In our exercise, we identify \( u \) as \( \sec 2\theta + \tan 2\theta \) and then find \( du \) as \( (\sec 2\theta \tan 2\theta + \sec^2 2\theta) d\theta \) making it possible to rewrite the integrand in terms of \( u \) and \( du \) alone. This method not only streamlines the integration process but also introduces a uniformity that allows for the application of basic integration rules.

Natural Logarithm in Integration

The natural logarithm arises naturally in the process of integrating functions of the form \( \frac{1}{u} \) following substitution. The integral \( \int \frac{1}{u} du \) simplifies to \( \ln |u| + C \) where \( C \) denotes the constant of integration.

Upon integrating our exercise’s simplified form \( \int \frac{1}{u} du \) we obtain \( \ln |u| + C \) as the antiderivative. To complete the integration process, we replace \( u \) with the original expression \( \sec 2\theta + \tan 2\theta \) to express our answer in terms of the original variable, which in this case is \( \theta \) rather than \( x \). Mastery of the natural logarithm in integration is essential for students as it frequently appears in solving rational functions and forms the backbone of many advanced calculus techniques.