Table of Contents
Basic concepts of Integration:
In Differential Calculus, we have learned to find the derivative/differential of a differentiable function. A natural question that may come to one’s mind is that what is the inverse method of differential calculus. More precisely, if we know the derivative of a function, then can we determine the function? Let’s understand this with an example.
Let
In the above example, given that
Integration is called the inverse method of differentiation.
What is Integration?
1. The method to find a function from its known differential is called integration.
2. The function we determine from the differential of a given function is called integral.
Definition and Notation of Integration:
Let
In symbolic notation, we write
Summary:
In other words,
Let’s understand the above facts with examples.
Example 1. Recall that
So by the definition of integration,
Example 2. We know that
Thus by definition,
Application of Integration:
The method of integration is generally used to find the area of a region bounded by curves.
List of all Integration Formulas
It is helpful to keep all integral formulas handy while solving problems of integrations. Here we list all integral formulas in one place.
Basic Integration Formulas:
1.
2.
3.
4.
5.
6.
7.
Trigonometric Functions Integration Formulas:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Also Read:
Integration Formulas (Substitution Method):
1.
2.
3.
4.
5.
6.
7.
8.
Some Special Integration Formulas:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Integration by Parts Formulas
To find the integration of the product of functions, we use the technique of integration by parts. If
The above formula is called the integration formula by parts. In this formula, we call the function
L -> logarithmic functions (example
I -> inverse functions (example
A -> algebraic functions (example x2)
T -> trigonometric functions (example
E -> exponential functions (example ex)
For example, if we have to evaluate
Properties of Definite Integrals
1. If
The above is the main property of definite integrals.
2.
3.
4.
5.
6.
7. If
8.
9.
10.
11.
12.
Few Examples
Now we will provide few examples with solutions to familiar the students with the above integral formulas.
Example 1:
Example 2:
Example 3:
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