# Integration: A Branch of Mathematics

The subject mathematics has been classified into various branches such as algebra, geometry, trigonometry, calculus, and many more. The concept of differentiation and integration comes under the branch of calculus. Differentiation can be defined as a process or method by which we can calculate the derivative of a given function. This formula is known as the differentiation formula. The most common example is velocity. It can be regarded as the rate of change in the position of a given object with respect to the time taken. The opposite of finding a derivative is known as the anti-differentiation formula. The opposite of differentiation is integration. It is the method by which we can calculate the surface area of the curve. It can also be defined as the method by finding the area under a curve. This area is found by decrypting the number of sides or faces of the polygon. Integration is also known as __integration by parts__. Both differentiation and integration are the sole concepts that shape the branch calculus. In this article, we shall cover some interesting topics such as methods of integration, integral by parts, and so on.

## Integrals

The antiderivatives of a function can be found with the help of the integrals. We have already discussed that the process of finding antiderivatives is known as integration. __Integrals__ can be defined as the inverse process or method of finding a derivative. The methods of integration are divided into 4 types. Such as the method of decomposition, method of substitution, method of partial fractions, and by solving parts. We shall discuss it in the next section.

## Various Methods of Integration

As mentioned above, integration is the process by which we can find the area under a curve. The methods of integration are divided into 4 types. Such as the method of decomposition, method of substitution, method of partial fractions, and by solving parts. The following are the various methods of integration.

- The process of integration by decomposing the sum or by differentiating the functions can be defined as the method of decomposition. The mentioned integrand will be in an algebraic, trigonometric, or exponential manner.
- The method of substituting integrals can be defined as the method of substitution. In this process, the variables of the integration can be easily changed and the function can be derivative.
- The integration by the parts was proposed by the mathematician Brook Taylor in 1715. This process is considered as an additional formula to calculate the antiderivative function. It can be used to integrate two or more two functions.

The above-mentioned methods are regarded as the additional methods of calculating the antiderivative of a function.

## What is Differentiation?

Differentiation is one of the methods by which you can find the derivative of a given or particular function. It is a process where we find the rate of change of a given object with respect to a different quantity. For example, the rate of change in quantity x with respect to quantity y can be defined as the process of differentiation. The concept of differentiation has been divided into various rules such as chain rule, product rule, quotient rule, the rule of sum and differences, and many more. The sum and difference rule of differentiation is used when the derivative of the function is a sum or difference. The chain rule helps to perform the differentiation of various functions which are defined as composite functions. Differentiation is also widely used in our everyday life and in various other fields such as physics, statistics, and so on. The following points are the applications of differentiation.

- The differentiation formula is used to find the rate of change for a given quantity. Such as velocity.
- It is also used to calculate the approximation value of a given quantity.
- Find the maximum and minimum values, the point where inflection occurs, and so on.
- The most basic use is to determine the increase of a function and decreasing of a function.

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