Problem 17 - Calculus 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

\(\int \operatorname{cosec}^{2} x d x=-\cot x+c\)

Step-by-Step Solution

Answer

The integral is \(-\cot x + C\).

Step 1: Identify the Integral

The exercise asks us to solve the integral \( \int \operatorname{cosec}^{2} x \, dx \). This integral involves the cosecant squared function, which is commonly seen in basic trigonometric integrals.

Step 2: Recall The Antiderivative

The antiderivative of \( \operatorname{cosec}^{2} x \) is a known result from integral calculus. Specifically, \( \int \operatorname{cosec}^{2} x \, dx = -\cot x + C \), where \( C \) is the constant of integration.

Step 3: Apply The Antiderivative

Apply the antiderivative to the integral. Therefore, the solution to the integral \( \int \operatorname{cosec}^{2} x \, dx \) is \(-\cot x + C\).

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \operatorname{cosec} \, x \), is one of the six fundamental trigonometric functions. It is the reciprocal of the sine function. So, \( \operatorname{cosec} \, x = \frac{1}{\sin \, x} \). This function is defined wherever the sine function is not equal to zero, ensuring that division by zero is avoided.
Understanding trigonometric functions like the cosecant is essential in solving integrals involving these functions.

• The cosecant function has specific properties, such as being undefined at the points where \( \sin \, x = 0 \), like at \( x = n\pi \) where \( n \) is any integer.
• It is periodic with a period of \( 2\pi \), meaning it repeats its pattern over intervals of \( 2\pi \).
• Graphically, the cosecant function appears as repeated vertical branches that grow indefinitely at angles where the sine function is zero.

Antiderivative

The antiderivative, often referred to as the "indefinite integral," is the inverse process of differentiation. It involves finding a function whose derivative results in the given function. In integral calculus, knowing the antiderivative can help solve integrals easily.
For instance, the antiderivative of the function \( \operatorname{cosec}^{2} x \) is \( -\cot x + C \). Here, \( C \) represents the constant of integration, which is crucial because it accounts for any constant that could have been part of the original function before differentiation.

• Understanding antiderivatives requires familiarity with basic integration rules and common functions.
• Memorizing key antiderivatives, such as that of \( \operatorname{cosec}^{2} x \), aids in efficiently solving integrals.

Being comfortable with finding antiderivatives opens up the ability to solve a wide range of mathematical problems involving calculus.

Integral Calculus

Integral calculus is a branch of mathematics focused on the processes of integration and its applications. It allows us to calculate areas under curves, among many other things, and is integral to understanding the behavior of functions.
Within integral calculus, we often work with the fundamental theorem, which connects differentiation and integration. This theorem states that the integral of a function's derivative over an interval returns the change in the function itself over that interval.
Key aspects of integral calculus include:

• Definite Integrals: These calculate the net area between a function and the x-axis over a specific interval. The result is a numerical value.
• Indefinite Integrals: These represent a family of functions rather than a single numeric value and include the constant of integration \( C \).
• The process of integration can be thought of as "gluing together" infinitesimally small pieces to find whole areas or quantities.

Mastering integral calculus concepts can enormously benefit students and professionals in fields that involve mathematical modeling and analysis.