Problem 11 - Calculus 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

\(\int \cos x d x=\sin x+c\)

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

The given problem is an integral calculus problem requiring us to evaluate the integral \( \int \cos x \, dx \). Our task is to find a function whose derivative is \( \cos x \).

Step 2: Set Up the Integral

The integral provided is \( \int \cos x \, dx \). This is a standard integral that can be solved using basic integration rules.

Step 3: Integration Rule

Recall that the derivative of \( \sin x \) is \( \cos x \). Therefore, when integrating \( \cos x \), the result is \( \sin x \) since differentiation and integration are inverse operations.

Step 4: Find the Integral

Using the integration rule, calculate the indefinite integral: \[ \int \cos x \, dx = \sin x + C \]. Here, \( C \) represents the constant of integration.

Step 5: Verify the Solution

To check the solution, differentiate \( \sin x + C \). The derivative of \( \sin x \) with respect to \( x \) is \( \cos x \), and the derivative of a constant \( C \) is zero. Therefore, \( \frac{d}{dx}(\sin x + C) = \cos x \), confirming our integration was correct.

Key Concepts

Indefinite Integral

The indefinite integral is a fundamental concept in calculus which represents a family of functions rather than a single number or function. An indefinite integral is noted by the integral symbol \( \int \) followed by the function to be integrated and usually expressed with respect to a variable, such as \( dx \). The result of integrating a function is another function plus an arbitrary constant, denoted by \( C \). This constant arises because differentiation eliminates constants, making them impossible to recover purely from the derivation process. For instance, when dealing with \( \int \cos x \, dx \), the integral can be expressed as \( \sin x + C \).

• The \( C \) represents any constant value.
• Indefinite integrals are also known as antiderivatives.
• Every continuous function has an indefinite integral.

Understanding indefinite integrals is essential for solving basic and advanced integration problems and plays a crucial role in the analysis of functions.

Basic Integration Rules

Basic integration rules serve as shortcuts to finding the antiderivatives of functions more efficiently. They are to integration what basic derivative rules are to differentiation, making complex calculus problems easier to tackle. These rules include several core concepts, one of which is the power rule, translating a familiar pattern for functions in the form of \( x^n \). For instance, the power rule for integration is given by:\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{(for } n eq -1\text{)}\]Other important rules include:

• Constant Rule: \( \int a \, dx = ax + C \)
• Sum/Difference Rule: \( \int (f(x) \pm g(x)) \, dx = \int f(x) \, dx \pm \int g(x) \, dx \)
• Trigonometric Rule: \( \int \cos x \, dx = \sin x + C \)

These rules help form the backbone of integral calculus, enabling the solving of numerous integral problems with function manipulations.

Derivative of Trigonometric Functions

To better understand integration, it's crucial to look at its inverse operation: differentiation. Specifically, the derivatives of trigonometric functions form a central part of calculus. Each trigonometric function has a unique derivative which couples it closely to another function, providing foundational elements for integration. For example, consider these derivative rules:

• The derivative of \( \sin x \) is \( \cos x \).
• The derivative of \( \cos x \) is \(-\sin x \).
• The derivative of \( \tan x \) is \( \sec^2 x \).

Understanding these derivatives allows you to work backwards to find their integrals because integration is the process of finding a function whose derivative matches a given function.In the exercise \( \int \cos x \, dx = \sin x + C \), knowledge of the derivative of \( \sin x \) being \( \cos x \) helps confirm that their relationship through integration is correct. Attention to these kinds of derivative patterns in trigonometry is not only helpful but also critical when approaching problems involving antiderivatives.