Integration: Learn Definition, Types, Formulas, Examples

We have learned the formulas for finding the area of a circle and ellipse. But how would you go about calculating the area of other curved shapes? This is where we need Integration. Integration can be used to find areas, volumes, central points, and many useful things. Most notably, it is used to find the area under the curve displayed below:

When it comes to calculus, Integration is one of the two major calculus topics in Mathematics (the other one being differentiation). Integration is a core part of the Mathematics syllabus for Classes 11 and 12. To know about Integration in detail, read the complete article:

Integration: What Is Integration?

Definition of Integration: Integration is the technique of finding a function g(x) the derivative of which, Dg(x), is equal to a given function f(x). This is indicated by the integral sign “∫,” as in ∫f(x), usually called the indefinite integral of the function.

The definite integral, is written as: y = ∫ f(x) dx, with a and b called the limits of integration, is equal to g(b) − g(a), where Dg(x) = f(x).

In addition to this, Mathematician Bernhard Riemann suggests that “Integral is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs.” This can be understood by looking at the image shown below:

Integration is understood as the summation of discrete data. The integral is calculated to find the functions which will describe the area, displacement, volume, that occurs due to a collection of small data, which cannot be measured separately.

It is crucial to understand that Integration is a type of calculus among its two types:

Differential Calculus
Integral Calculus

In application within Mathematics, integration is used for:

Finding the problem function, when its derivatives are given.
Calculating the area bounded by the graph of a function under certain constraints.

Integration: Types of Integrals

In Mathematics, there are two types of Integrals:

1. Definite Integral

An integral that contains the upper and lower limits then it is a definite integral. Definite Integral is also called a Riemann Integral when it is restricted to lie on the real line. It is represented as:

2. Indefinite Integral

Indefinite integrals are defined without upper and lower limits. An indefinite integral is represented as:

Integration: List of Formulas For Integration

Students practicing integration require a constant reference to formulas to learn integration. Even if you have grasped the concepts of Integration, it is crucial to refer to formulas as a helpful resource.

∫ 1 dx = x + C
∫ a dx = ax+ C
∫ xⁿdx = ((xⁿ⁺¹)/(n+1))+C ; n≠1
∫ sin x dx = – cos x + C
∫ cos x dx = sin x + C
∫ sec²dx = tan x + C
∫ csc²dx = -cot x + C
∫ sec x (tan x) dx = sec x + C
∫ csc x ( cot x) dx = – csc x + C
∫ (1/x) dx = ln |x| + C
∫ e^xdx = e^x+ C
∫ a^xdx = (a^x/ln a) + C ; a>0, a≠1

Solved Examples Using Integration Formula

Example 1: Solve ∫(x² + 3x – 2)dx

∫(x²+ 3x – 2)dx
= x³/3+(3x²)/2 – 2x + c

Example 2: Solve ∫ (1/x⁷) dx

= ∫ x^-7 dx
= x^{(-7 + 1)}/(-7 + 1) + c
= x^-6/(-6) + c
= (-1/6x⁶) + c

Example 3: Solve ∫ (cos x / sin² x) dx

∫(cos x / sin² x) dx
=  ∫(cosx/sinx) (1/ sinx) dx
=  ∫cot x cosec x dx
= – cosec x + c

Example 4: Integrate the following with respect to x – ∫ (x²⁴/x²⁵) dx

∫ (x²⁴/x²⁵) dx =  ∫ x^24-25 dx
=  ∫ x^-1 dx
=  ∫ (1/x) dx
= log x + c

Example 5: ∫ (1 – x²)^-1/2 dx

∫ (1 – x²)^-1/2dx = ∫ 1/(1 – x²)^1/2dx
= ∫ 1/√(1 – x²)dx
= sin^-1 x + c

Frequently Asked Questions About Integration

Let’s look at some of the commonly asked questions about Integration:

Question: What is Integration?
Answer: Integration is defined as the technique of finding a function g(x) the derivative of which, Dg(x), is equal to a given function f(x). This is indicated by the integral sign “∫,” as in ∫f(x), usually called the indefinite integral of the function.

Question: What is integration used for?
Answer: Integration is used for finding area of abstract and curvy objects in real life. Conceptually, you start by finding area under the curve, and then use the same knowledge to plot area of more complex objects. It is also used to find distance, velocity, acceleration, volume, center of mass, average value of a function, probability, arc length, etc.

Question: What are the two different types of integrals?
Answer: The two different types of integrals are definite integral and indefinite integral.

Question: What are some of the commonly used formulas in Integration?
Answer: Some of the commonly used formulas in integration are:
∫ 1 dx = x + C
∫ a dx = ax+ C
∫ xⁿdx = ((xⁿ⁺¹)/(n+1))+C ; n≠1
∫ sin x dx = – cos x + C
∫ cos x dx = sin x + C
∫ sec²dx = tan x + C
∫ csc²dx = -cot x + C
∫ sec x (tan x) dx = sec x + C
∫ csc x ( cot x) dx = – csc x + C
∫ (1/x) dx = ln |x| + C
∫ e^xdx = e^x+ C
∫ a^xdx = (a^x/ln a) + C ; a>0, a≠1

Question: What is the difference between integration and differentiation?
Answer: While differentiation is used to study changes happening in smaller quantities with respect to unit change in other variables, Integration is used to add multiple small and discrete values which cannot be added singularly.

We hope you found all the information related to Integration helpful. Integration is one of the very important chapters within the JEE Main Syllabus. To improve your knowledge in Mathematics, you can take the JEE Main Chapter-wise Tests as well.