Problem 41 - Calculus Single Variable

Question

Use trigonometric identities to compute the indefinite integrals. $$ \int \frac{\sin (x)}{\cos ^{2}(x)} d x $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Trigonometric Identity

The given integral is \( \int \frac{\sin(x)}{\cos^2(x)} \, dx \). Recognize that \( \frac{\sin(x)}{\cos^2(x)} \) can be rewritten using the identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) and \( \sec(x) = \frac{1}{\cos(x)} \). Therefore, \( \frac{\sin(x)}{\cos^2(x)} = \tan(x) \cdot \sec(x) \).

Step 2: Rewrite the Integral

Rewrite the integral using the identity found: \( \int \frac{\sin(x)}{\cos^2(x)} \, dx = \int \tan(x) \cdot \sec(x) \, dx \).

Step 3: Determine the Antiderivative

Use the substitution method here. Let \( u = \sec(x) \) so that \( du = \sec(x) \tan(x) \, dx \). This means \( \int \tan(x) \cdot \sec(x) \, dx = \int du \), as \( du \) directly matches the integrand. Thus, the integral becomes \( \int du \).

Step 4: Integrate with Respect to \( u \)

Since \( \int du \) is simply \( u + C \), we substitute back \( u = \sec(x) \) to obtain \( \sec(x) + C \).

Step 5: Write the Final Answer

Thus, the indefinite integral of the given function is \( \sec(x) + C \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables where both sides of the equation are defined. They are essential in simplifying complex trigonometric expressions. In the context of indefinite integrals, they allow us to transform an integrable function into a simpler form, which is easier to integrate.

• Tangent and Secant Identities: Two key identities used in integration are the tangent identity, \( an(x) = \frac{\sin(x)}{\cos(x)} \), and the secant identity, \( \sec(x) = \frac{1}{\cos(x)} \). Combining these, you can transform fractions with trigonometric functions, like \( \frac{\sin(x)}{\cos^2(x)} \), into a product form such as \( \tan(x) \cdot \sec(x) \), which can simplify the integration process.

Recognizing and applying these identities is a key skill in solving trigonometric integrals. It helps break down complex looking expressions into manageable forms, using the relations between sine, cosine, tangent, and secant.

Antiderivative

The antiderivative, also known as the indefinite integral, is a function whose derivative is the original function given. Finding an antiderivative is like solving for the reverse of a derivative: it tells us the family of functions whose slope matches the given function.

• General Indefinite Integral Form: The indefinite integral of a function \( f(x) \) is represented as \( \int f(x) \, dx = F(x) + C \), where \( F'(x) = f(x) \) and \( C \) is the constant of integration. This constant signifies that there are many functions with the same derivative, differing only by a constant.

In our solution, after simplifying the integral using trigonometric identities, we found an antiderivative by using the substitution method. The result was \( \sec(x) + C \), which indicates all functions with this formula will have the derivative \( \frac{\sin(x)}{\cos^2(x)} \). Remember, accurately identifying and solving for the antiderivative is crucial for proper integration.

Substitution Method

The substitution method is a pivotal technique in calculus for solving integrals, especially when integrating complex expressions. The essence of substitution is to transform an integral into a simpler form that is easier to handle.

• Choosing the "u": The first step is to select a substitution \( u \) whose derivative \( du \) matches part of the original integrand. In our problem, choosing \( u = \sec(x) \) effectively simplified the integration process, as its derivative, \( du = \sec(x) \tan(x) \, dx \), matched the integrand.
• Simplifying the Integral: By substituting \( u \), the original integral \( \int \tan(x) \cdot \sec(x) \, dx \) becomes \( \int du \), an integral in terms of the new variable \( u \). This is far simpler to solve than the original expression.
• Reintegration back to x: Once the integral in terms of \( u \) is solved, substituting back to terms of \( x \) finalizes the indefinite integral. From our example, we returned to \( \sec(x) + C \).

Mastering the substitution method allows for efficient solutions to otherwise daunting integration problems. It ensures the integral is aligned with recognizable formulas and simplifies the overall process.