Problem 332 - Calculus Volume 1

Question

In the following exercises, find each indefinite integral by using appropriate substitutions. $$\int \tan \theta d \theta$$

Step-by-Step Solution

Answer

The indefinite integral is \(-\ln |\cos \theta| + C\).

Step 1: Rewrite the Integral Using Trigonometric Identity

Recognize that \( \tan \theta \) can be expressed as \( \frac{\sin \theta}{\cos \theta} \). Thus, rewrite the integral as:\[ \int \tan \theta \, d\theta = \int \frac{\sin \theta}{\cos \theta} \, d\theta. \]

Step 2: Choose a Suitable Substitution

To integrate \( \frac{\sin \theta}{\cos \theta} \), let \( u = \cos \theta \). Then the derivative \( du = -\sin \theta \, d\theta \). Solving for \( \sin \theta \, d\theta \), we get \( -du = \sin \theta \, d\theta \).

Step 3: Substitute and Simplify

Substitute \( u = \cos \theta \) and \( -du = \sin \theta \, d\theta \) into the integral:\[ \int \frac{\sin \theta}{\cos \theta} \, d\theta = \int \frac{-1}{u} \, du. \]

Step 4: Integrate

The integral of \( \frac{-1}{u} \) with respect to \( u \) is:\[ -\ln |u| + C, \]where \( C \) is the constant of integration.

Step 5: Back-Substitute to Original Variable

Replace \( u \) with \( \cos \theta \) to write the final answer in terms of \( \theta \):\[ -\ln |\cos \theta| + C. \]

Step 6: Final Step: Write the Indefinite Integral Solution

The indefinite integral \( \int \tan \theta \, d\theta \) is computed to be:\[ \int \tan \theta \, d\theta = -\ln |\cos \theta| + C, \]where \( C \) is the constant of integration.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a valuable technique in calculus for simplifying the process of integrating functions that contain trigonometric expressions. In this context, we use trigonometric substitution to replace certain expressions with trigonometric identities that are easier to integrate or differentiate. For example, when you encounter an integral involving a trigonometric function like \( \tan \theta \), rewriting it in terms of \( \sin \theta \) and \( \cos \theta \) can simplify the integration process. This transforms the problem into one involving basic trigonometric identities that are easier to handle.Substitution also involves selecting a new variable in place of a function component, such as setting \( u = \cos \theta \) in our example. This not only makes the integral simpler but also allows us to use well-known integration formulas, reducing the complexity step by step. With practice, trigonometric substitution can be mastered and applied to a variety of integral problems.

Integration Techniques

Integration techniques are strategies applied to determine the integral of a function. There are several techniques to choose from, and selecting the right one can simplify the problem.

• For basic functions, the power rule or simple antiderivatives provide quick solutions.
• For more complex trigonometric integrals, methods like substitution can significantly simplify the process.
• Other techniques include integration by parts, partial fraction decomposition, and numerical integration methods.

In our example, the method of substitution is introduced by letting \( u = \cos \theta \), which transforms the integral into a form more familiar and solvable. By substituting, we temporarily change the perspective of the problem to simplify the mathematical manipulation. This technique is particularly effective when dealing with products or quotients of trigonometric functions.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are accurate for various angles. These identities are crucial for both simplifying and solving mathematical expressions involving trigonometric functions.The identity used in the exercise, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), is just one of many that simplify expressions. Understanding these identities allows us to break down complex trigonometric functions into simpler forms, making them easier to work with.

• Common identities include Pythagorean identities, angle sum and difference identities, and double angle formulas.
• Rewriting expressions using these identities can reveal integrals in a form that is straightforward to solve.

By mastering trigonometric identities, students can become much more adept in calculus, particularly in solving integrals that involve trigonometric functions. Recognizing which identity to use is the key skill that grows with practice.