Problem 20 - Biocalculus Calculus for the Life Sciences

Question

Evaluate the indefinite integral. \(\int \sec 2 \theta \tan 2 \theta d \theta\)

Step-by-Step Solution

Answer

\( \frac{1}{2} \sec(2\theta) + C \)

Step 1: Identify the Integral

The integral given is \( \int \sec 2\theta \tan 2\theta \, d\theta \). We need to evaluate this indefinite integral.

Step 2: Consider a U-substitution

A common method to solve integrals involving trigonometric functions is substitution. Notice that if we let \( u = 2\theta \), then \( du = 2 \, d\theta \). Rearranging gives \( d\theta = \frac{1}{2} \, du \).

Step 3: Substitute into the Integral

Substitute \( u = 2\theta \) and \( d\theta = \frac{1}{2} \, du \) into the integral: \[\int \sec u \tan u \cdot \frac{1}{2} \, du = \frac{1}{2} \int \sec u \tan u \, du.\]

Step 4: Evaluate the New Integral

The integral \( \int \sec u \tan u \, du \) is a standard integral. It is known that: \[ \int \sec u \tan u \, du = \sec u + C. \] Therefore, \[ \frac{1}{2} \int \sec u \tan u \, du = \frac{1}{2}(\sec u) + C. \]

Step 5: Substitute Back the Original Variable

Recall that we let \( u = 2\theta \). Substitute back to get the result in terms of \( \theta \): \[ \frac{1}{2}(\sec(2\theta)) + C. \]

Step 6: Write the Final Answer

The indefinite integral is thus evaluated as: \[ \int \sec 2\theta \tan 2\theta \, d\theta = \frac{1}{2} \sec(2\theta) + C, \] where \( C \) is the constant of integration.

Key Concepts

Indefinite Integrals

When we talk about integrals, we're usually referring to either definite or indefinite integrals. Indefinite integrals are unique because they don't have limits attached; they represent a family of functions rather than a single value.
An indefinite integral is written using the integral sign (∫) and includes a function and a variable of integration. The result of an indefinite integral is a function plus a constant of integration, typically denoted as \( C \), because integrating a function can lead to multiple possible answers that only differ by a constant.
Indefinite integrals can be thought of as finding the original function when only the derivative is known. For example, if you're given a function's rate of change, integrating it can tell you what the original function was, except for a constant. This makes indefinite integrals a powerful tool for uncovering the original functions hidden within their derivatives.
In the given exercise, the task was to evaluate the indefinite integral \( \int \sec 2\theta \tan 2\theta \, d\theta \), leading up to the form of a function involving \( \theta \) and a constant \( C \).

U-Substitution

U-substitution is a clever technique used to simplify the process of integrating functions, especially when dealing with trigonometric integrals. The idea is to transform a complex integral into a more manageable form.
Essentially, you choose a substitution \( u \) to replace a part of the integral, which simplifies the expression. As in the exercise, if we let \( u = 2\theta \), then the derivative \( du = 2 \, d\theta \). Rearranging gives \( d\theta = \frac{1}{2} \, du \). This substitution helps in changing the variable of integration from \( \theta \) to \( u \), making the integral easier to compute.
After you've completed the integration process, you substitute back to the original variable to find the final answer. In this exercise, substituting back \( u = 2\theta \) allowed us to express the integral in terms of \( \theta \) again. U-Substitution is a fundamental technique, making complex integrals more approachable and often necessary to solve them.

Integration Techniques

Integration techniques are critical tools in solving integrals, as they provide strategies to tackle different forms of integrals. Each technique is like a key designed to unlock specific types of integral doors.
Some common integration techniques include substitution (like u-substitution), integration by parts, trigonometric substitution, and partial fraction decomposition. Each technique has its own set of rules and applications that make certain integrals easier to solve.
In the case of the integral \( \int \sec 2\theta \tan 2\theta \, d\theta \), u-substitution proved to be effective. It transformed a more complex problem into one with a recognizable solution: \( \int \sec u \tan u \, du \). This is a standard trigonometric integral with a known result, \( \sec u + C \).
Mastering these integration techniques is crucial, as it sharpens your problem-solving toolkit, enabling you to tackle a wide variety of integrals with confidence. Understanding when and how to apply each method can greatly simplify the integration process.