Integration

• In mathematics, when we cannot perform general addition operations, we use integration to add values on a large scale.

• There are various methods in mathematics to integrate functions.

• Just like, addition- subtraction, multiplication-division integration, and differentiation are also a pair of inverse functions.

• According to integration definition math, it is a process of finding functions whose derivative is given is named anti-differentiation or integration.

• Integration is a process of adding slices to find the whole.

• It can be used to find areas, volumes, and central points.

 What is Integration in Maths?

• According to integration definition maths to find the whole, we generally add or sum up many parts to find the whole. 

• As we know, Integration is a reverse process of differentiation, which is a process where we reduce the functions into smaller parts.

• To find the summation under a very large scale the process of integration is used.

• We can use calculators for the calculation of small addition problems which is a very easy task to do. In problems where the limits reach infinity, we use integration methods, to sum up, many parts. 

• Differentiation and Integration both are important parts of calculus. 

In Mathematics, we Know That There are two Major Types of Calculus –

1. Differential Calculus

2. Integral Calculus

What do you mean by Integral Calculus?

Let us now talk about Integral Calculus,

• Integral Calculus is the branch of calculus concerned with the application of integrals.

• To see what differential calculus is, let us take an example of a slope of a line in a graph. In any graph, we can find the slope of a line, using the slope formula. What do we do when we need to find the slope of a curve?

The slope of the points in a curve varies and here differential calculus comes into the picture.

Different Types of Integrals in Mathematics-

Types of Integration-

Till now we have learned what Integration is. In mathematics, there are two types of Integrations or integrals -

• Definite Integral

• Indefinite Integral

What is a Definite Integral?

• A definite integral is an integral that contains both the upper and the lower limits. 

• Definite Integral is also known as Riemann Integral.

Integration is a method of adding or summing up the parts to find the whole. It is just a reverse process of how differentiation is calculated, where we reduce the various functions into small parts. This method is used to find the summation where there are a huge amount of calculations involved. 

Along with the concept of differentiation, integration is a basic and fundamental but very essential concept of calculus and serves as a very informative tool to solve problems mainly in mathematics and physics field involving the calculation of area of an arbitrary shape, the length of a curve, the slope of a curve, and area under the curve, and also the volume of a solid, among others. 

Formulas of Integration

• ∫ xn.dx = x(n + 1)/(n + 1)+ C.

• ∫ 1.dx = x + C.

• ∫ ex.dx = ex + C.

• ∫1/x.dx = log|x| + C.

• ∫ ax.dx = ax /loga+ C.

• ∫ ex(f(x) + f'(x)).dx = ex.f(x) + C.

In addition to the method of substitution, which is already familiar in mathematics, there are three basic principal methods to solve an integration which will be studied in this chapter are, reduction to trigonometric integrals, decomposition into partial fractions, and integration by parts.

An integral number is usually assigned to functions in a way that describes and involves measurement of displacement, area, volume, and other concepts that arise by combining infinitesimal data. This process of finding the integrals is called integration in math.

If f = x, and Dg = cos x, then ∫x·cos x = x·sin x − ∫sin x = x·sin x − cos x + C. 

Integrals are used to evaluate some specific quantities such as area, volume, work, and, in general, any quantity that can be interpreted as the area under a curve.

In Math, integration is a method of adding or summing up the different individual parts into a whole. It is just a whole reverse process of differentiation, where we reduce a function into various different parts. This method is used to find the summation under a vast scale.