Integration - Definition, Types, Methods, Examples & FAQs

Table of Contents

What is Integration?

Integration is the process of combining two or more things into a single entity. In business, integration is the process of combining two or more companies into a single entity. This can be done through a merger, acquisition, or joint venture. Integration – Definition Types Methods Examples & FAQs.

Points to Remember:

1. The first step is to identify the problem.

2. Next, brainstorm possible solutions.

3. Select the best solution and implement it.

4. Evaluate the solution to determine if it was successful.

5. If the solution was not successful, brainstorm more possible solutions.

6. Select the best solution and implement it.

7. Evaluate the solution to determine if it was successful.

8. Repeat steps 5-7 until the problem is solved.

1. The first step is to identify the problem.

2. Next, brainstorm possible solutions.

3. Select the best solution and implement it.

4. Evaluate the solution to determine if it was successful.

5. If the solution was not successful, brainstorm more possible solutions.

6. Select the best solution and implement it.

7. Evaluate the solution to determine if it was successful.

8. Repeat steps 5-7 until the problem is solved.

Types of Integration Maths or the Integration Techniques-

There are many types of integration maths techniques. The most commonly used techniques are:

1. Rectangular Method
2. Trapezoidal Method
3. Simpson’s 1/3 Rule
4. Newton’s Method

For better understanding here’s what each method is!

1. Integration by Substitution –

Integrate the function

using the substitution

The result is

2. Integration by Parts –

Integration by Parts is a technique used to integrate a function of two variables. The technique splits the function into two functions, one of which is a function of a single variable, and the other of which is a product of the function and a simple constant. The technique is then applied to each of the two functions, and the integrals are combined to give the final integral.

Integration by Parts Example

Find the integral:

The integral can be split into two functions, one of which is a function of a single variable, and the other of which is a product of the function and a simple constant.

The function can be split into:

and

The integral can be applied to each of the two functions.

and

The integrals can be combined to give the final integral.

The final integral can be evaluated.

What is the LIATE Rule?

The LIATE Rule is an acronym for the four main types of tenses in English:

L – past tense
I – present tense
A – future tense
T – present perfect tense

3. Integration Using Trigonometric Identities –

We can use trigonometric identities to simplify integrals.

For example, the integral

can be written as

Now we can use the trigonometric identity

to simplify the integral.

We get

which can be easily evaluated.

4. Integration of Some Particular Function –

The particular function that is being integrated is the function

, where is a constant.

The integration is performed with respect to the variable .

The integral is

where is a constant.

5. Integration by Partial Fraction –

Inverse

Let’s say we have the function:

$f(x) = \frac{x}{1+x}$

We can integrate this function using the standard method, but we can also integrate it using partial fractions.

To do this, we first need to find the inverse of the function. This can be done using the following equation:

$x = \frac{1+x}{x}$

Once we have the inverse, we can use it to find the partial fractions of the original function.

For our example, the inverse is:

$x = \frac{1+x}{x}$

We can now use this inverse to find the partial fractions of the original function.

For our example, the partial fractions are:

$\frac{1}{1+x} + \frac{x}{1+x}$

We can now integrate the function using these partial fractions.

For our example, the integral is:

$\frac{1}{1+x} + \frac{x}{1+x} \cdot \frac{dx}{1+x}$

We can simplify this integral by using the substitution $u = 1+x$.

For our example, the integral is:

$\frac{du}{1+x} + \frac{x}{1+x}

Topics Covered in Methods of Integration: Definitions, Types, Examples

Types of Integration

Definite Integral

Indefinite Integral

Types of Integration

There are two main types of integration: definite and indefinite.

Definite Integral

A definite integral calculates the area between two curves. It is a measure of the accumulation of a quantity (such as distance, volume, or mass) over a period of time.

To calculate a definite integral, you need to know the starting point and the end point of the area you are measuring. You also need to know the function that describes the two curves.