Pure mathematics

1. Integration
Suppose your friend gives you a wooden stick. He asks you to break it. Can you do
so? Yes, it will be very easy for you to do so. But what will happen if he gives you
five to six sticks to break? It will not be that easy to break it. As the number of sticks
increases it is difficult to break them. The process of uniting things is an integration of
things. Similarly, in mathematics too, we have an integration of two functions.
Integration is like drop by drop addition of water in a container
Integration Definition
The integration denotes 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 singularly. In a
broad sense, in calculus, the idea of limit is used where algebra and geometry are
implemented. Limits help us in the study of the result of points on a graph such as
how they get closer to each other until their distance is almost zero. We know that
there are two major types of calculus –
 Differential Calculus
 Integral Calculus
Maths Integration:
In Maths, integration is a method of adding or summing up the parts to find the
whole. It is a reverse process of differentiation, where we reduce the functions
into parts, this process is the reverse of finding a derivative. Integrations are the anti-
derivatives. Integrations are the way of adding the parts to find the whole. Integration
is the whole pizza and the slices are the differentiable functions which can be
integrated. If f(x) is any function and f′(x) is its derivatives. The integration of f′(x)
with respect to dx is given as:
∫ f′(x) dx = f(x) + C

2. (The symbol for "Integral" is a stylish "S" that looks like the example above)
Integration is divided into 2 types:
1. Definite Integral: An integral of a function with limits of integration. There
are two values as the limits for the interval of integration. One is the lower
limit and the other is the upper limit. It does not contain any constant of
integration.
2. Indefinite Integral: It is an integral of a function when there is no limit for
integration. It contains an arbitrary constant.
What is the Constant of Integral?
The constant of integration expresses a sense of ambiguity. For a given derivative
there can exist many integrands which may differ by a set of real numbers. This set of
real numbers is represented by the constant, C.
Basic Rules of Integration:
Examples:
 ∫ 𝟗. 𝒅𝒙
Sol = 9x + c
 ∫ 𝟏𝟎. 𝒅𝒙
Sol = 10x + c
 ∫ 𝟐𝟖. 𝒅𝒙
Sol = 28 + c
𝟏: ∫ 𝒙. 𝒅𝒙 = 𝒂𝒙 + 𝒄

3. Examples:
 ∫ 𝒙𝟖
. 𝒅𝒙
Sol =
𝒙𝟖+𝟏
𝟖+𝟏
+ 𝒄 =
𝒙𝟗
𝟗
+ 𝒄
 ∫ 𝒙𝟔
. 𝒅𝒙
Sol =
𝒙𝟔+𝟏
𝟔+𝟏
+ c =
𝒙𝟕
𝟕
+ 𝒄
 ∫ 𝒙−𝟑
. 𝒅𝒙
Sol =
𝒙−𝟑+𝟏
− 𝟑+𝟏
+ c =
𝒙−𝟐
−𝟐
+ 𝒄
 ∫ 𝒙
𝟏
𝟐 . 𝒅𝒙
Sol =
𝒙
𝟏
𝟐
+𝟏
𝟏
𝟐
+𝟏
+c =
𝒙
𝟐
𝟑
𝟑
𝟐
+ c
𝟐: ∫ 𝒙𝒏
. dx =
𝒙𝒏+𝟏
𝒏+𝟏
+ 𝒄 (n ≠ -1)

4. Examples:
 ∫ 𝟓𝒙−𝟑
. 𝒅𝒙
Sol =
𝟓𝒙−𝟑+𝟏
𝟑+𝟏
+ c =
𝟓𝒙−𝟐
−𝟐
+ 𝒄
 ∫ 𝟏𝟐𝒙𝟐
. 𝒅𝒙
Sol =
𝟏𝟐𝒙𝟐+𝟏
𝟐+𝟏
+ 𝒄 =
𝟏𝟐 𝒙𝟑
𝟑
+ 𝒄
 ∫ √𝒙 + 𝟑𝒙𝟐
. 𝒅𝒙
Sol = ∫ 𝒙
𝟏
𝟐 . 𝒅𝒙 + ∫ 𝟑𝒙𝟐
. 𝒅𝒙
=
𝒙
𝟐
𝟑
𝟐
𝟑
+ 𝒙𝟑
+ 𝒄
 ∫ 3𝒙𝟐
+
𝟒
𝒙𝟐 . 𝒅𝒕
Sol = ∫ 𝟑𝒙𝟐
+ ∫
𝟒
𝒙𝟐 . 𝒅𝒕
= 3 ∫ 𝒙𝟐
. 𝒅𝒕 + 𝟒∫ 𝒙−𝟐
. 𝒅𝒕
= 3
𝒙𝟑
𝟑
+ 𝟒
𝒙−𝟏
−𝟏
= 𝒙𝟑
−
𝟒
𝒙
+ 𝒄
𝟑: ∫ 𝒂𝒙𝒏
. dx =
𝒙𝒏+𝟏
𝒏+𝟏
+ 𝒄 (n ≠ -1)