Problem 55 - Thomas Calculus

Question

In Exercises \(17-56,\) find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int \cos \theta(\tan \theta+\sec \theta) d \theta$$

Step-by-Step Solution

Answer

The antiderivative is \(-\cos \theta + \theta + C\).

Step 1: Expand the Expression

First, expand the expression inside the integral by distributing \( \cos \theta \) across \( (\tan \theta + \sec \theta) \). This results in two separate terms: \( \cos \theta \tan \theta + \cos \theta \sec \theta \).

Step 2: Simplify Each Term

Recall the trigonometric identities: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \). Applying these, the expression simplifies to \( \sin \theta + 1 \).

Step 3: Integrate Term by Term

Find the antiderivative of each term separately: the antiderivative of \( \sin \theta \) is \( -\cos \theta \) and the antiderivative of \( 1 \) is \( \theta \). Thus, the overall antiderivative is \( -\cos \theta + \theta + C \), where \( C \) is the constant of integration.

Step 4: Verify by Differentiation

Differentiate the result \( -\cos \theta + \theta + C \) to check if it equals the original integrand. The derivative is \( \sin \theta + 1 \), which indeed matches the simplified form of the integrand, confirming the solution is correct.

Key Concepts

Antiderivative

When we find the antiderivative, also known as the indefinite integral, we look for a function whose derivative is the given function. It involves reversing the process of differentiation. In this exercise, the goal is to find a function whose derivative is the same as the expression inside the integral. If you take the derivative of the function you found, it should match the original function.
When you're working with basic functions like polynomials, this might be easy. However, with more complex expressions, like trigonometric ones, it might require a bit of manipulation. This is where expanding and simplifying expressions helps in identifying the antiderivative.

Trigonometric Identities

Trigonometric identities are fundamental in simplifying trig expressions when solving integrals. They provide relationships between different trigonometric functions that can simplify complex expressions.
For example, in the problem, we use the identities \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \) to simplify \( \cos \theta (\tan \theta + \sec \theta) \) into a simpler form, \( \sin \theta + 1 \). This step turns a challenging integral into something more straightforward.
Remember, different problems might require different identities, and knowing these can help in transforming an expression into a more workable form.

Differentiation

Differentiation is the reverse process of finding an antiderivative. It's used to verify if the function we found as an antiderivative is correct.
In context, after finding the antiderivative \( -\cos \theta + \theta + C \), we differentiate it to ensure that we obtain back the function \( \sin \theta + 1 \). This serves as a checkpoint for our solution. The power of differentiation lies in its ability to find the rate of change or derivative of a function. When we differentiate our solution, we expect it to return to the original integrand, confirming our process and answer are correct.

Constant of Integration

The constant of integration \( C \) is a vital component of the antiderivative. When integrating, it accounts for the infinite number of vertical shifts that an indefinite integral might have. Since differentiation removes constants, when you find an antiderivative, you're left with a family of functions that differ by a constant. As seen here, the antiderivative result is \( -\cos \theta + \theta + C \). The inclusion of \( C \) is crucial because it reflects all possible versions of the antiderivative that differ by a constant. Without this constant, you'd have an incomplete solution to the indefinite integral.