Integral is categorically a branch of mathematics which is derived from calculus. Before defining the integral, we just want to give you a brief introduction about mathematics and its branches.

Mathematics is the study of numbers, their relations and related operations. Mathematics is used in almost all types of fields such as medicine, social sciences & engineering. It has widely been extended in the field of physical science & technology. So we can say that the math is everywhere.

There are so many braches of mathematics which covers the entire world of science as an essential tool to deals with the quantitative calculations and analytical or logical reasoning problems. The principle branches of mathematics are as below:

- Algebra
- Arithmetic
- Calculus
- Geometry
- Trigonometry
- Numerical Analysis
- Optimization
- Probability
- Set Theory
- Topology
- Euclidean
- Non Euclidean

To discuss all branches in a single topic is not possible because all branches have their specific features and also categorized into further sub branches. In this topic, we will only discuss about the calculus. Calculus is sub divided into two categories:

- Differential Calculus
- ii). Integral Calculus

Our main concern in this topic is to familiarize yourself about integral by definition, describing and exploring it with different examples.

Integral refers to the problem that creates while linking infinitesimal data to find the values of displacement, area, volume etc. The term integral is also used as definite integral whereas indefinite integral describes antiderivatives. The following symbol is used for integral:

$$\int\limits$$Definite integral is defined as a function in which integral has lower and upper limits and represented as

$$∫_a^bf(x) dx$$Indefinite integral is defined as a function in which integral has no lower and upper limits and represented as

$$∫f(x) dx = f(x) + c$$Where x is the real number and c is content.

The procedure of finding the integral values is called integration. Basically, integration refers to the process of combining smaller scattered units into a single one.

The graph can easily explain the following definition of integral:

“To find the displacement, volume, area and other concepts, numbers are assigns to functions in a specific way by combining infinitesimal data is called integral”

Integrals have different properties that are used to solve the problems associated with it. Moreover, antiderivative calculator is also available with the capability of solving the complex integral or anti-derivative problems.

Formulas to solve the indefinite integral problems

A set of basic integral formulas is available to solve the indefinite integral problems which are given below:

- $$∫a dx=ax+c$$
- $$∫1 dx = x + c$$
- $$∫sin x dx = - cos x + c$$
- $$∫cosec² x dx = - cot x + c$$
- $$∫cosec x (cot x) dx = - cosec x + c$$
- $$∫cos x dx = sin x + c$$
- $$∫sec ²x dx = tan x + c$$
- $$∫sec x (tan x) dx = sec x + c$$
- $$∫〖(1/x)〗 x dx = ln IxI + c$$
- $$∫e² dx = ex + c$$
- $$∫ax dx = (ax/ln a) + c (where a ≠ 1, a >0)$$
- $$∫xn dx = ((xn+1) / n + 1)) + c (where n ≠ 1) $$ (Also called Power Rule)

Example 1:

Solve $$∫x^3 dx$$ by using Power Rule?

Solution:

Given that $$∫x^3 dx$$

We know the power rule of integral or integration

$$∫x^n dx = {{x^(n+1) \over n+1} + c}$$

Put the value of n=3

$$∫x^3 dx = {x^(3+1)\over3+1} + c$$

$$∫x^3 dx = {x^4\over4} + c$$

Example 2:

Calculate the indefinite integral

$$∫(6x^2 – 10x + 2cosx) dx $$

Solution:

$$(6x^2 – 10x + 2cosx) dx$$

$$= ∫6x^2 dx - ∫10x dx + ∫2cosx dx$$

$$= 6∫x^2 dx - 10∫x dx + 2∫cosx dx$$

Evaluate all integral by using integration table

$$I = 6.(x³ )/3 - 10.(x²)/2 + 2.sinx + C$$

$$= 2x^3 – 5x^2 + 2.sinx + C$$

There are some formulas that are followed to solve the definite integral problems. These are also called the properties of definite integrals:

- $$∫_a^a f(x) dx = 0$$
- $$∫_a^b cf(x) dx = c∫_a^bf(x) dx$$
- $$∫_a^b [f(x) - g (x) ] dx = ∫_a^b f(x) dx - ∫_a^b g(x) dx $$ (Also called Difference Rule)
- $$∫_a^b [f(x) + g (x) ] dx = ∫_a^b f(x) dx + ∫_a^b g(x) dx $$ (Also called Sum Rule)
- $$If f (x) ≥ g (x) on [a , b], then ∫_a^b f(x) dx ≥ ∫_a^b g(x) dx$$
- $$If f (x) ≤ 0 on [a , b], then ∫_a^b f(x) dx ≤ 0$$
- $$∫_a^b f(x) dx = -∫_b^af(x) dx$$
- $$∫_a^bcdx = c (b – a) ( C is Constant)$$
- $$∫_a^b f(x) dx = ∫_a^c f(x) dx + ∫_c^b f(x) dx $$(when there are three points on a closed interval)
- $$∫_a^b f(x) dx = f (c)(b - a) $$(Mean Value Theorem)

Example 1:

Evaluate $$∫_0^5 -6x^2 dx, if ∫_0^5x^2 dx = 11$$
Solution:

Given values are

$$a = 0$$
$$b = 5$$
$$f (x) = -6$$
$$∫_0^5x^2 dx = 11$$
Put the known values in the given formula

$$∫_a^b f(x) dx = -∫_a^b f(x) dx$$
$$∫_0^5 -6x^2 dx = -6∫_0^5x2 dx$$
$$∫_0^5 -6x^2 dx = (-6) (11)$$
$$∫_0^5 -6x^2 dx = -66$$

Example 2:

Determine $$∫_2^{10} 5 dx$$
Solution:

Given is $$∫_2^{10} 5 dx$$
Identify the values

$$a = 2$$
$$b = 10$$
$$c = 5$$
Put the values in the formula

$$∫_a^b cdx = c (b – a) $$( C is Constant)
$$∫_5^{10} 5 dx =5(10-2)$$
Simplify the equation

$$∫_5^{10} 5 dx =5(8)$$
$$∫_5^{10} 5 dx =40$$

Example 3: Determine $$∫_2^8 f(x) dx if ∫_2^{12} f(x) dx = 15 and ∫_8^{12} f(x) = 10$$
Solution:

Given that

$$∫_2^{12} f(x) dx = 15$$
$$∫_8^{12} f(x) = 10$$
$$∫_2^8 f(x) dx = ?$$
We know the formula to simplify the integral

$$∫_a^b f(x) dx = ∫_a^c f(x) dx + ∫_c^b f(x) dx$$
We can substitute the values as

$$a = 2$$
$$b = 12$$
$$c = 8$$
Put the values in the formula

$$∫_a^b f(x) dx = ∫_a^c f(x) dx + ∫_c^b f(x) dx$$
$$∫_2^{12} f(x) dx = ∫_2^8 f(x) dx + ∫_8^{12} f(x) dx$$
Re-arrange the formula

$$∫_2^8 f(x) dx = ∫_2^{12} f(x) dx - ∫_8^{12} f(x) dx$$
$$∫_2^8 f(x) dx = 15 - 10$$
$$∫_2^8 f(x) dx = 5$$

Verify the above result of this definite integral by entering the function in integral calculator.

Integrals are used in many fields such as engineering, physics, finance & economics to find the mass & momentum of a tower, center of gravity, area under the curve, volume, and velocity over distance etc. It is also used by the graphic designers in making three dimensional models.