Problem 26 - Calculus: Early Transcendental Functions

Question

Find the general antiderivative. $$\int \frac{\cos x}{\sin x} d x$$

Step-by-Step Solution

Answer

The general antiderivative of the given function is \(-\ln|\sin x| + C\)

Step 1: Recognize the Integrable Function

First, recognize the integrand. Here, \(\frac{\cos x}{\sin x}\) trigonometric function is given to integrate. This integrand can be transformed into a more recognizable form by using the property that \(\frac{\cos x}{\sin x}\) is equivalent to \(\cot x\). Thus, the integral becomes: \(\int \cot x dx\)

Step 2: Apply the Antiderivative of Cotangent

The next step is to apply the basic formula for the antiderivative of the cotangent function. The antiderivative or integral of \(\cot x\) is \(-\ln|\sin x|\). Hence, the integral is: \(-\ln|\sin x| + C\), where C is the constant of integration.

Key Concepts

Trigonometric Integrals

Trigonometric integrals are special types of integrals involving trigonometric functions like sine, cosine, tangent, etc. In this exercise, we're dealing with \(\frac{\cos x}{\sin x}\), which can be simplified to a more standard form for integration. By recognizing that \(\frac{\cos x}{\sin x}\) is equivalent to \(\cot x\), you can easily turn the complex-looking ratio into a simpler trigonometric integral. Identifying these transformations is key for solving trigonometric integrals efficiently.

• Breaking down \(\frac{\cos x}{\sin x}\) to \(\cot x\) helps simplify the integration process.
• Trigonometric identities are crucial: in our case, recognizing a simple trigonometric function was key.

Understanding these concepts makes handling trigonometric integrals less intimidating, providing a clearer path to finding the antiderivatives.

Calculus Techniques

Solving integrals often involves a variety of calculus techniques that enable you to transform the given problems into more solvable forms. In the case of trigonometric functions, using substitution or recognizing identities is a common practice. For this integral, recognizing the identity \(\frac{\cos x}{\sin x} = \cot x\) was crucial.

• Transformation Technique: Transform the integrand by simplifying using trigonometric identities.
• Integration by Recognition: Spot known integrals, like the integral of \(\cot x\), which has a straightforward formula.

Mastering these techniques enhances your ability to tackle a wide range of integrals, including those involving trigonometric functions. The core idea is flexibility in using different approaches depending on what the problem requires.

Integration Formulas

Using integration formulas can greatly simplify the process of finding antiderivatives. These formulas serve as shortcuts that allow you to quickly solve integrals without re-deriving them every time. In this exercise, the integral of \(\cot x\) is solved using the known formula for its antiderivative.

• The Integral of Cotangent: \(-\ln|\sin x|\), which is a result you can find in calculus formula tables.
• Constant of Integration: Remember to add \(C\), the constant of integration, because antiderivatives are not unique; they differ by a constant.

Leveraging these well-established formulas helps streamline the integration process, particularly for standard functions like \(\cot x\). Knowing these by heart can speed up solving exercises and aid in tackling more complex problems efficiently.