Problem 4 - Calculus Single Variable

Question

Calculate the indefinite integral. $$ \int \sec (x) \tan (x) d x $$

Step-by-Step Solution

Answer

The integral is \( \sec(x) + C \).

Step 1: Recognize the Integral Form

The integral given is \( \int \sec(x) \tan(x) \, dx \). This is a standard form of an indefinite integral that involves the trigonometric functions \( \sec(x) \) and \( \tan(x) \).

Step 2: Recall the Basic Antiderivative

Recall the basic antiderivative: \( \frac{d}{dx}[ \sec(x) ] = \sec(x) \tan(x) \). This implies that the integral of \( \sec(x) \tan(x) \) is simply \( \sec(x) \).

Step 3: Write Down the Antiderivative Result

Since the derivative of \( \sec(x) \) is \( \sec(x) \tan(x) \), the antiderivative of \( \sec(x) \tan(x) \) is \( \sec(x) \).

Step 4: Add the Constant of Integration

For indefinite integrals, always remember to add the constant of integration \( C \). So the complete antiderivative is \( \int \sec(x) \tan(x) \, dx = \sec(x) + C \).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental elements in mathematics, especially in calculus. They include functions like sine ( sin (x)), cosine ( cos (x)), and tangent ( tan (x)). In this exercise, we are focusing on the secant ( sec (x)) and tangent ( tan (x)). These functions are important because they help us understand relationships between angles and sides of triangles.

In the context of integration, recognizing these functions and their derivatives can simplify solving problems. For instance, the derivative of sec (x) is sec (x) tan (x). Knowing this simplifies finding antiderivatives, because the reverse operation, integration, becomes straightforward when you can match the function and its derivative pattern.

Antiderivative

An antiderivative, also known as an indefinite integral, is a function whose derivative is the original function you started with. In our exercise, we identified that \(int sec(x) tan(x) dx\) simplifies to sec(x) because the derivative of sec(x) gives us sec(x)tan(x).

Finding antiderivatives is a fundamental process in calculus that involves reversing differentiation. Here's the basic process:

• Identify the function whose derivative matches your integrand.
• Use known antiderivative formulas or rules that match the integrand's form.
• Verify by differentiation that your result is correct.

Understanding antiderivatives is key to solving complex calculus problems, as they allow you to determine cumulative sum functions from their rates of change.

Constant of Integration

The constant of integration, denoted as C , appears in the solution of indefinite integrals. When you compute an indefinite integral, you are essentially finding all possible antiderivatives of a function.

Since differentiation of a constant yields zero, different functions can have the same derivative if they only differ by a constant value. This means:

• Whenever you integrate, you must add C to represent all potential vertical shifts along the y-axis.
• The constant of integration accounts for the fact that integration is essentially finding a family of functions.
• Remembering to include C is crucial in capturing the full set of solutions an integral can represent.

Including this constant ensures that we correctly represent all antiderivatives, acknowledging an infinite set of possible functions beyond a singular solution.