Problem 21 - Analysis 1: Ein Lehr- und Arbeitsbuch für Studienanfänger

Question

Bestimimen sie $$ \int \frac{1}{\sin x} d x $$

Step-by-Step Solution

Answer

The integral equals \( \ln |\csc x - \cot x| + C \).

Step 1: Rewrite the Integral

First, observe that \( \sin x = \frac{1}{\csc x} \). So the integral \( \int \frac{1}{\sin x} \, dx = \int \csc x \, dx \).

Step 2: Find an Antiderivative of \( \csc x \)

The antiderivative of \( \csc x \) is a standard result which is \( \ln |\csc x - \cot x| + C \).

Step 3: Write the General Solution

Thus, the integral \( \int \csc x \, dx \) equals \( \ln |\csc x - \cot x| + C \), where \( C \) is the constant of integration.

Key Concepts

Integral Calculus

Integral calculus is a fundamental concept in mathematics and is primarily concerned with the idea of integration. Integration is the process of finding a function relative to its derivative, essentially the reverse of differentiation. This technique allows us to calculate areas under curves, volumes of shapes, and more.

In the context of integration, there is a distinction between definite and indefinite integrals. Definite integrals calculate the area under a curve between two points, giving a numerical value. In contrast, indefinite integrals determine the antiderivative of a function, leading to a general expression plus a constant of integration. Understanding these concepts is key in many areas of mathematics and science.

Antiderivatives

An antiderivative is a function whose derivative is the original function we started with. It is essential for solving many calculus problems, especially in integration. When you integrate a function, you are essentially finding its antiderivative.

For example, when integrating a trigonometric function like \( rac{1}{ ext{sin}(x)} \), you are looking for a function whose derivative is \( rac{1}{ ext{sin}(x)} \). The solution to this is expressed in terms of natural logarithms as \( ext{ln} | ext{csc}(x) - ext{cot}(x)| \), where \( C \) is the constant of integration. This constant arises because an indefinite integral accounts for all potential vertical shifts of the antiderivative, making it a family of functions.

Trigonometric Integrals

Trigonometric integrals involve integrating functions that include trigonometric expressions such as sine, cosine, tangent, and others. These integrals often require advanced techniques or identities to solve.

The integral \( \int \frac{1}{\sin x} \, dx \) is an example of a trigonometric integral. To solve it, one converts \( \frac{1}{\sin x} \) into \( \csc x \), recognizing the csc (cosecant) as the reciprocal of sine. The antiderivative of \( \csc x \) is known from standard calculus results, enabling us to express the solution in terms of the natural logarithm: \( \ln |\csc x - \cot x| + C \). Understanding these expressions and identities is crucial for successful problem solving in calculus.

Mathematical Problem Solving

Mathematical problem solving involves using a variety of methods and approaches to overcome challenges and find solutions to mathematical problems. Integral calculus, trigonometric identities, and antiderivatives are tools frequently used in this process.

Solving the problem \( \int \frac{1}{\sin x} \, dx \) demonstrates several steps in mathematical problem solving. It involves recognizing trigonometric identities, rewriting functions to simplify integration, and using known results for antiderivatives. These skills are fundamental for tackling more complex functions and are essential for students learning calculus.

Effective problem solving often includes:

• Understanding the problem.
• Identifying useful identities or rules.
• Applying step-by-step logical reasoning.
• Rechecking results to verify accuracy.

These steps ensure a comprehensive and thorough approach to solving mathematical exercises.