Integrals

Integrals

Edited By Team Careers360 | Updated on May 07, 2022 12:55 PM IST

Finding a derivative is the inverse of this procedure. Integrations are the polar opposite of derivatives. Integrations are a method of putting the pieces together to find the whole. The whole pizza represents integration, and the pieces are the differentiable functions that can be integrated. If f(x) is any function and f′(x) is the derivative of that function. With regard to dx, the integration of f′(x) is given as

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∫ f′(x) dx = f(x) + C . Because of the rapid growth in the automation technology industry, system integration is becoming increasingly important. Also, because of the resulting need to streamline procedures for better control and management. An integrated technique will modernise operations while lowering costs and maintaining efficiency. The integrals are divided into two types. Indefinite Integrals: When there is no limit to the integration of a function, it is called an indefinite integral. It has an arbitrary constant in it. Definite Integrals are integrals of functions having integration limitations. The interval of integration has two values as its boundaries. The lower limit is one, while the top limit is the other. It doesn't have any integration constants. The integration constant conveys a sense of uncertainty. There can be several integrands for a given derivative, each differing by a set of real numbers. The constant C represents this collection of real numbers.

List of topics according to NCERT and JEE Main/NEET syllabus:

Introduction to integration
Indefinite integration
Theorems based on the integration
Properties of definite integration
Limit of sum
Integration by partial fraction
Integration by parts
Integration of some functions

Important concepts and Laws:

The term "integration" refers to the process of putting things together. It connects and completes slices. However, we'll use it to calculate the area of any random curve in this case. If I ask you to calculate the area of this curve, you might not be able to do so. But what if we break this curve down into smaller chunks? This should make things easier. But what if we break it down even further? Wouldn't that be much better? Similarly, reducing the size of the pieces will cause them to assume the shape of the specified curve, resulting in a much more accurate computed area. But, with everything being so little, how can we add things up now? We don't have to, after all. There is another approach to calculate this area, which requires just finding the antiderivative of the provided function. Integration is the inverse of Differentiation, as you may have guessed. As a result, we know that we must write to calculate the derivative of f(x).

?/??(?(?))=?(?)

Theorems based on the integration

Differentiation and integration are the polar opposites of each other.
The sum of two integrands' integration is the sum of two integrands' integrations.
For any real number k, ∫ k f(x) dx = k ∫ f(x) dx.

It's a technique for simplifying a system of equations by expressing one variable in terms of another and thereby eliminating one variable from the equation. Then, calculate this equation and back replace it till you get the solution. The substitution approach aims to simplify one of the equations by rewriting it in terms of a single variable. Furthermore, the crucial point to remember is that you are constantly swapping equal values.

Follow these steps to solve a system of equations:

To begin, solve one equation for one variable (e.g., in y = or x = form).
Substitute this expression for the missing variable in the other equation and solve.
Then, to acquire an answer, replace your solution into the first equation and solve it.

Some Special Integrals

∫dx/(x² + y²) = 1/y tan^-1 x/y + K
∫dx/(x² - y²) = 1/2y * log |x - y/x + y| + K
∫dx/(y² - x²) = 1/2y * log |y + x/y - x| + K
∫dx/√(y² - x²) = Sin^-1 x/y + K
∫dx/√(x² + y²) = log |x + √(x² + y²)| + K

NCERT Notes Subject wise link:

Importance of Integral

In terms of your final board test as well as other competitive examinations, Chapter 7 of Class 12 Maths is a crucial chapter. The notion of integration, which encompasses integral calculus and its characteristics, is explained in this chapter. This Chapter contains a number of essential questions that stem from important concepts and themes. Integral evaluation, determination, functions of integrals, anti-derivatives, and numerous integration methods are among them. You must continue to practice in order to do well in Chapter 7 of Class 12 Maths. You'll need a personal notebook to keep track of all your notes. This will assist you in remembering the concepts and subjects you have studied. This will assist you in completely preparing for the exam and ensuring that you remember all of the crucial concepts and subjects you learned. It will provide you with more information and a better grasp of the topic. Making notes will also assist you in your editing process.

NCERT Solutions Subject wise link:

NCERT Exemplar Solutions Subject wise link:

Frequently Asked Question (FAQs)

1. What is the purpose of the integration?

Integration is used to measure things from a very fundamental and common standpoint. It might be a measurement of length, area, or volume. Probabilities can also be used in the context of unplanned variables. It aids in the evaluation of a function's behavior. In whatever precise meaning in which one wishes to assess conduct.

2. What are the benefits of integrating systems?

Horizontal integration may be beneficial to businesses, and it usually occurs when they are competing in the same industry or area. These benefits include optimizing business processes, reducing costs, expanding market share, reducing competition, and achieving economies of scale.

3. What does it mean to have an indefinite integral?

A function that takes the antiderivative of another function is referred to as an infinite integral. In other words, infinite integrals do not have lower and upper limits. We depict it graphically as an integral symbol, a function, and finally a dx.

4. Why is the area under the curve the antiderivative?

When a function f(x) is integrated, you obtain antiderivative F(x). You may also acquire the area under the curve by evaluating the anti-derivative over a given domain [a, b].

5. Can you tell the difference between an indefinite and a definite integral?

The top and lower boundaries of a definite integral define it. Furthermore, it is referred to as definitive since it delivers a definite solution at the conclusion of the problem. Indefinite integral, on the other hand, refers to a more comprehensive type of integration. Furthermore, the anti-derivative of the considered function is the interpretation of the indefinite integral