

Integration
Integration is a method of bringing different parts to form a whole. Integration is simply the inverse process of differentiation. We use integration in mathematics to find areas, volumes, etc. The area of the region enclosed by the graph of functions is defined and calculated using integration.
Integration Rules of Basic Functions
We have some integration rules to find out the integral of more complicated functions than just the known ones. These rules can be read in the sections below. Besides these rules, there are a number of integral formulas that can be used to replace the integral form.
Different types of functions have different integration rules. Let's look at some basic integration rules for some basic functions, such as:
1. Constant Function
Integration of constant function say ‘a’ will result in:
For example:
Where C is the integral constant.
2. Linear Variable Function
If x is any given variable then, we can write this as,
Where C is the integral constant.
3. Square Function
If the square function is given, then,
Where C is the integral constant.
4. Reciprocal Function
We know 1/x is a reciprocal function of x. The integration for this function can be written as:
Where C is the integral constant.
5. Exponential Function
The rules of integration of different exponential functions are given below:
Where C is the integral constant.
6. Trigonometric Function
Where C is the integral constant.
Important Integration Rules
As we discussed basic integration rules above, let’s also look at some of the important rules for integration below:
Power Rule
Sum Rule
Different Rule
Multiplication by Constant
Power Rule
If we integrate x raised to the power n according to the power rule of integration, we get:
Where C is the integral constant.
The integration of a square function is also derived from this rule, i.e.
Example: Integrate
Where C is the integral constant.
Sum Rule
According to the sum rule, the integration of the sum of two functions equals the sum of their individual integrals.
Example:
Where C and D are the integration constants. So the integration can be written as:
Now K is the integration constant here.
Difference Rule
The integration difference rule is identical to the sum rule.
Example:
Where C and D are the integration constants. So the integration can be written as:
Now K is the integration constant here.
Multiplication by Constant
When a function is multiplied by a constant, the integration of that function is as follows:
Example:
Where C is the integral constant.
Methods of Integration
In some cases, visual inspection is insufficient to determine the integral of a function. There are other ways to obtain the integral of a function that is reduced to its standard form. Other methods are mentioned below:
Decomposition method
Integration using Partial Fractions
Integration by Parts
Decomposition Method
In this method we decompose the complicated function in the form of sum or difference of simple functions. For example, 2sinA.cosB can be decomposed as
Integration by Substitution
This method is known as reverse chain rule method and u-substitution. In this method we transform a complicated integral to a standard form using substitution.
For example,
So, here we substitute
Re-substitute,
Where C is integral constant.
Integration using Partial Fractions
If a rational function has a complicated denominator then this method will be used to decompose and integrate the function.
For example,
Integration by Parts
The product rule of integration is another name for this rule. When two functions are multiplied together, then this integration method is used.
The formula for integration by parts is:
Here,
u is a function of u(x)
v is a function of v(x)
u’ will be the derivative of the function u(x)
Here we select u(first function) and v(second function) according to the ILATE rule.
L - Logarithm Functions
A - Algebraic Functions
E - Exponential Functions
Did You know?
If the derivative of a function is given and asked to find the integration of it then we can simply write the function as answer. For example, 2x is derivative of
Solved Examples
1.
Ans:
Where c is integral constant.
2.
Ans:
Here we substitute
Therefore
And the above equation becomes
Putting the value of u in the equation we get
Where c is the integral constant.
Practice Questions
Integrate the following functions:
Conclusion
Integration is the process of determining the antiderivatives of functions, often called integrals. There are numerous approaches to integrating a function. The basic integration formulas are used to find the antiderivatives for a few standard integrals. There are several methods to use, including substitution, by parts integration, and partial fractions integration.
FAQs on Do You Know The Important Integration Rules?
1. What are the four integral rules?
The four integral rules are:
Power Rule
Sum Rule
Difference Rule
Multiplication by Constant
2. What is the integral of 1?
Sometimes students get confused between differentiation and integration. Here 1 is constant.
Differentiation of any constant is 0 but the integration of any constant is given as:
So,
Where c is the integral constant.
3. What is integration used for?
Integration is the term used to describe the process of finding integrals. Integration, like differentiation, is a fundamental, essential operation of calculus that can be used to answer issues in math and physics involving the area of an arbitrary form, the length of a curve, and the volume of a solid, among other things.

















