Integral calculus is a fundamental sub-type of mathematics that is used to study definite and indefinite integrals. The area under the curve between two boundary points will be evaluated with the help of a definite integral.

While the indefinite integral is used to evaluate the antiderivative of a function. The definite and indefinite integrals are frequently used in many fields such as physics, engineering, economics, etc.

In this article, we’ll cover the basics of definite and indefinite integrals with the help of examples. 

What is integral calculus?

In calculus, a mathematical concept that is used to evaluate the antiderivative of a function and the area under the curve among two points is said to be integral. The way of finding and evaluating the area under the curve and antiderivatives of function is known as integration.

 The integral can be imagined as the sum of infinitely small rectangular areas under a curve. The area of each rectangle is equal to the value of the function at that point multiplied by the width of the rectangle.

By summing up these infinitely small rectangles, we can find the area under the curve. It is extensively used in different fields of science. There are two types of integration that are helpful in finding the area under the curve and the antiderivative of a function.

• Definite integral
• Indefinite integral

What is the definite integral?

The definite integral is a kind of integration that is used to evaluate the area under the curve with the help of interval points of the function. The limits of integration are specified in this sub-type of integral calculus.

The expression for the definite integral can be written as:

∫^y_x f(u) du

Where, x & y are the specified limits of integration, f(u) is the function and du is the integrating variable of the function.

The formula expression of the definite integral can be evaluated with the help of the fundamental theorem of calculus. It states that “the definite integral of a function f(u) between two interval points x & y is equivalent to the difference between the antiderivative of f(u) evaluated at “y” and the antiderivative of f(u) evaluated at “x””. Such as

∫yx f(u) du = [F(u)]yx F(u) is the antiderivative of the function with respect to “u”.

According to the fundamental theorem ∫yx f(u) du = F(y) – F(x)

What is the indefinite integral?

In calculus, the indefinite integral is a way of determining a new function (original function) whose integrand is equal to the derivative. In this subtype of integral, the limits are not specified. The process of evaluating a new function (indefinite integral) is said to be the antiderivative of a function.

The expression of the indefinite integral will be written without upper and lower limits such as:

∫ f(u) du

Where f(u) is a differential function that is to be integrated to find the antiderivative. The constant of integration must be written along with the antiderivative of a function. such as:

∫ f(u) du = F(u) + C

F(u) is the antiderivative of the function with respect to “u”. C is the constant of integration.

Relation between Area under the curve and Antiderivative

The calculation of areas under the curve and antiderivative can be done with the help of definite and indefinite integrals respectively. Both types of integrals involve integrating a function that’s why they are closely related.

The fundamental theorem of calculus is a way of showing the relation between the definite and indefinite integrals as first we have to find the antiderivative of a function and then by placing the interval point, we get the area under the curve.

How to calculate the definite and indefinite integrals?

The calculations of the definite and indefinite integral will be explained with the help of examples. Here are a few solved examples to understand the concept of evaluating the area under the curve and finding the antiderivative of a function.

Example 1

Find the antiderivative of the given function with respect to “u”

f(u) = 5sin(u)– 12u³ + 4u + 3u² – 50 

Solution

Step 1:Write the given function according to the general expression of the indefinite integral.

ʃ f(u) du = ʃ [5sin(u)– 12u³ + 4u + 3u² – 50] du

Step 2:Now use the properties of the integral calculus to write the notation of integration with each term separately.

ʃ [5sin(u)– 12u³ + 4u + 3u² – 50] du = ʃ [5sin(u)] du– ʃ [12u³] du + ʃ [4u] du + ʃ [3u²] du – ʃ [50] du

Step 3: Now take out the constant coefficients outside the integral notation to make each term standard.

ʃ [5sin(u)– 12u³ + 4u + 3u² – 50] du = 5ʃ [sin(u)] du– 12ʃ [u³] du + 4ʃ [u] du + 3ʃ [u²] du – ʃ [50] du

Step 4: Find the antiderivative with the help of power and trigonometry rules.

ʃ [5sin(u)– 12u³ + 4u + 3u² – 50] du = 5 [-cos(u)]– 12 [u³⁺¹ / 3 + 1] + 4 [u¹⁺¹ / 1 + 1] + 3 [u²⁺¹ / 2 + 1] – [50u] + C

ʃ [5sin(u)– 12u³ + 4u + 3u² – 50] du = 5 [-cos(u)]– 12 [u⁴ / 4] + 4 [u² / 2] + 3 [u³ / 3] – [50u] + C

ʃ [5sin(u)– 12u³ + 4u + 3u² – 50] du = 5 [-cos(u)]– 12/4 [u⁴] + 4/2 [u²] + 3/3 [u³] – [50u] + C

ʃ [5sin(u)– 12u³ + 4u + 3u² – 50] du = 5 [-cos(u)]– 3 [u⁴] + 2/1 [u²] + 1 [u³] – [50u] + C

ʃ [5sin(u)– 12u³ + 4u + 3u² – 50] du = -5cos(u)– 3u⁴ + 2u² + u³ – 50u + C

An online integral calculator is an alternate way to find the result of the above example without involving in lengthy calculations.

Example 2

Calculate the area under the curve between the interval of [2, 3] with respect to “v”

f(v) = 2v⁵ + 3v² + 5v³ – v + 20

Solution

Step 1:Write the given function according to the general expression of the definite integral.

ʃ_x^y f(v) dv = ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv

Step 2:Now use the properties of the integral calculus to write the notation of integration with each term separately.

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = ʃ₂³ [2v⁵] dv+ ʃ₂³ [3v²] dv + ʃ₂³ [5v³] dv – ʃ₂³ [v] dv + ʃ₂³ [20] dv

Step 3: Now take out the constant coefficients outside the integral notation to make each term standard.

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 2ʃ₂³ [v⁵] dv+ 3ʃ₂³ [v²] dv + 5ʃ₂³ [v³] dv – ʃ₂³ [v] dv + ʃ₂³ [20] dv

Step 4: Find the antiderivative with the help of power and trigonometry rules.

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 2 [v⁵⁺¹ / 5 + 1]₂³ + 3 [v²⁺¹ / 2 + 1]₂³ + 5 [v³⁺¹ / 3 + 1]₂³ – [v¹⁺¹ / 1 + 1]₂³ + [20v]₂³

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 2 [v⁶ / 6]₂³ + 3 [v³ / 3]₂³ + 5 [v⁴ / 4]₂³ – [v² / 2]₂³ + 20 [v]₂³

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 2/6 [v⁶]₂³ + 3/3 [v³]₂³ + 5/4 [v⁴]₂³ – 1/2 [v²]₂³ + 20 [v]₂³

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 1/3 [v⁶]₂³ + [v³]₂³ + 5/4 [v⁴]₂³ – 1/2 [v²]₂³ + 20 [v]₂³

Step 5:Now find the area under the curve by putting the interval points to the antiderivative of the function.

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 1/3 [3⁶ – 2⁶]+ [3³ – 2³] + 5/4 [3⁴ – 2⁴] – 1/2 [3² – 2²] + 20 [3 - 2]

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 1/3 [729 – 64]+ [27 – 8] + 5/4 [81 – 16] – 1/2 [9 – 4] + 20 [3 - 2]

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 1/3 [665]+ [19] + 5/4 [65] – 1/2 [5] + 20 [1]

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 665/3+ 19 + 325/4 – 5/2 + 20

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 221.67+ 19 + 325/4 – 5/2 + 20

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 221.67+ 19 + 81.25 – 2.5 + 20

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 339.42

Final Words

The area under the curve and antiderivative is to key terms in calculus that are evaluated with the help of definite and indefinite integrals. The examples of these kinds of integrals will able you to solve any problem of finding antiderivative and calculating the area under the curve.

Integral calculus is a fundamental sub-type of mathematics that is used to study definite and indefinite integrals. The area under the curve between two boundary points will be evaluated with the help of a definite integral.

While the indefinite integral is used to evaluate the antiderivative of a function. The definite and indefinite integrals are frequently used in many fields such as physics, engineering, economics, etc.

In this article, we’ll cover the basics of definite and indefinite integrals with the help of examples. 

What is integral calculus?

In calculus, a mathematical concept that is used to evaluate the antiderivative of a function and the area under the curve among two points is said to be integral. The way of finding and evaluating the area under the curve and antiderivatives of function is known as integration.

 The integral can be imagined as the sum of infinitely small rectangular areas under a curve. The area of each rectangle is equal to the value of the function at that point multiplied by the width of the rectangle.

By summing up these infinitely small rectangles, we can find the area under the curve. It is extensively used in different fields of science. There are two types of integration that are helpful in finding the area under the curve and the antiderivative of a function.

• Definite integral
• Indefinite integral

What is the definite integral?

The definite integral is a kind of integration that is used to evaluate the area under the curve with the help of interval points of the function. The limits of integration are specified in this sub-type of integral calculus.

The expression for the definite integral can be written as:

∫^y_x f(u) du

Where, x & y are the specified limits of integration, f(u) is the function and du is the integrating variable of the function.

The formula expression of the definite integral can be evaluated with the help of the fundamental theorem of calculus. It states that “the definite integral of a function f(u) between two interval points x & y is equivalent to the difference between the antiderivative of f(u) evaluated at “y” and the antiderivative of f(u) evaluated at “x””. Such as

∫yx f(u) du = [F(u)]yx F(u) is the antiderivative of the function with respect to “u”.

According to the fundamental theorem ∫yx f(u) du = F(y) – F(x)

What is the indefinite integral?

In calculus, the indefinite integral is a way of determining a new function (original function) whose integrand is equal to the derivative. In this subtype of integral, the limits are not specified. The process of evaluating a new function (indefinite integral) is said to be the antiderivative of a function.

The expression of the indefinite integral will be written without upper and lower limits such as:

∫ f(u) du

Where f(u) is a differential function that is to be integrated to find the antiderivative. The constant of integration must be written along with the antiderivative of a function. such as:

∫ f(u) du = F(u) + C

F(u) is the antiderivative of the function with respect to “u”. C is the constant of integration.

Relation between Area under the curve and Antiderivative

The calculation of areas under the curve and antiderivative can be done with the help of definite and indefinite integrals respectively. Both types of integrals involve integrating a function that’s why they are closely related.

The fundamental theorem of calculus is a way of showing the relation between the definite and indefinite integrals as first we have to find the antiderivative of a function and then by placing the interval point, we get the area under the curve.

How to calculate the definite and indefinite integrals?

The calculations of the definite and indefinite integral will be explained with the help of examples. Here are a few solved examples to understand the concept of evaluating the area under the curve and finding the antiderivative of a function.

Example 1

Find the antiderivative of the given function with respect to “u”

f(u) = 5sin(u)– 12u³ + 4u + 3u² – 50 

Solution

Step 1:Write the given function according to the general expression of the indefinite integral.

ʃ f(u) du = ʃ [5sin(u)– 12u³ + 4u + 3u² – 50] du

Step 2:Now use the properties of the integral calculus to write the notation of integration with each term separately.

ʃ [5sin(u)– 12u³ + 4u + 3u² – 50] du = ʃ [5sin(u)] du– ʃ [12u³] du + ʃ [4u] du + ʃ [3u²] du – ʃ [50] du

Step 3: Now take out the constant coefficients outside the integral notation to make each term standard.

ʃ [5sin(u)– 12u³ + 4u + 3u² – 50] du = 5ʃ [sin(u)] du– 12ʃ [u³] du + 4ʃ [u] du + 3ʃ [u²] du – ʃ [50] du

Step 4: Find the antiderivative with the help of power and trigonometry rules.

ʃ [5sin(u)– 12u³ + 4u + 3u² – 50] du = 5 [-cos(u)]– 12 [u³⁺¹ / 3 + 1] + 4 [u¹⁺¹ / 1 + 1] + 3 [u²⁺¹ / 2 + 1] – [50u] + C

ʃ [5sin(u)– 12u³ + 4u + 3u² – 50] du = 5 [-cos(u)]– 12 [u⁴ / 4] + 4 [u² / 2] + 3 [u³ / 3] – [50u] + C

ʃ [5sin(u)– 12u³ + 4u + 3u² – 50] du = 5 [-cos(u)]– 12/4 [u⁴] + 4/2 [u²] + 3/3 [u³] – [50u] + C

ʃ [5sin(u)– 12u³ + 4u + 3u² – 50] du = 5 [-cos(u)]– 3 [u⁴] + 2/1 [u²] + 1 [u³] – [50u] + C

ʃ [5sin(u)– 12u³ + 4u + 3u² – 50] du = -5cos(u)– 3u⁴ + 2u² + u³ – 50u + C

An online integral calculator is an alternate way to find the result of the above example without involving in lengthy calculations.

Example 2

Calculate the area under the curve between the interval of [2, 3] with respect to “v”

f(v) = 2v⁵ + 3v² + 5v³ – v + 20

Solution

Step 1:Write the given function according to the general expression of the definite integral.

ʃ_x^y f(v) dv = ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv

Step 2:Now use the properties of the integral calculus to write the notation of integration with each term separately.

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = ʃ₂³ [2v⁵] dv+ ʃ₂³ [3v²] dv + ʃ₂³ [5v³] dv – ʃ₂³ [v] dv + ʃ₂³ [20] dv

Step 3: Now take out the constant coefficients outside the integral notation to make each term standard.

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 2ʃ₂³ [v⁵] dv+ 3ʃ₂³ [v²] dv + 5ʃ₂³ [v³] dv – ʃ₂³ [v] dv + ʃ₂³ [20] dv

Step 4: Find the antiderivative with the help of power and trigonometry rules.

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 2 [v⁵⁺¹ / 5 + 1]₂³ + 3 [v²⁺¹ / 2 + 1]₂³ + 5 [v³⁺¹ / 3 + 1]₂³ – [v¹⁺¹ / 1 + 1]₂³ + [20v]₂³

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 2 [v⁶ / 6]₂³ + 3 [v³ / 3]₂³ + 5 [v⁴ / 4]₂³ – [v² / 2]₂³ + 20 [v]₂³

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 2/6 [v⁶]₂³ + 3/3 [v³]₂³ + 5/4 [v⁴]₂³ – 1/2 [v²]₂³ + 20 [v]₂³

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 1/3 [v⁶]₂³ + [v³]₂³ + 5/4 [v⁴]₂³ – 1/2 [v²]₂³ + 20 [v]₂³

Step 5:Now find the area under the curve by putting the interval points to the antiderivative of the function.

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 1/3 [3⁶ – 2⁶]+ [3³ – 2³] + 5/4 [3⁴ – 2⁴] – 1/2 [3² – 2²] + 20 [3 - 2]

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 1/3 [729 – 64]+ [27 – 8] + 5/4 [81 – 16] – 1/2 [9 – 4] + 20 [3 - 2]

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 1/3 [665]+ [19] + 5/4 [65] – 1/2 [5] + 20 [1]

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 665/3+ 19 + 325/4 – 5/2 + 20

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 221.67+ 19 + 325/4 – 5/2 + 20

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 221.67+ 19 + 81.25 – 2.5 + 20

ʃ₂³ [2v⁵ + 3v² + 5v³ – v + 20] dv = 339.42

Final Words

The area under the curve and antiderivative is to key terms in calculus that are evaluated with the help of definite and indefinite integrals. The examples of these kinds of integrals will able you to solve any problem of finding antiderivative and calculating the area under the curve.

Two's complement is a mathematical operation that transforms a binary number into its negative counterpart. It is based on the concept of complementing bits, where 0s are changed to 1s, and 1s are changed to 0s. However, in two's complement, an additional step is performed after the complementation: adding one to the resulting number.

One must flip all the bits of a positive integer's positive counterpart before adding one to the result to represent a negative number using the two's complement. This makes it possible to express both positive and negative numbers in the same binary system consistently and distinctively.

In this article, we will discuss the definition of Two’s complement, solving methods, and properties of two’s complement. Also, the two’s complement will explain with the help of examples.

Definition of Two's complement

Two's complement is a binary representation scheme for signed integers where the most significant bit represents the sign, and negative numbers are obtained by inverting and adding one to the absolute value. The two's complement system is based on the concept of complementing and adding, and it offers several advantages over other methods of signed number representation.

Methods to find Two’s complement

To find the two's complements of a binary number, follow these steps:

• Start with the number's binary representation.
• The two's complement is identical to the initial binary representation if the integer is positive (the leftmost bit is 0).

Follow these procedures if the number is negative and the leftmost bit is 1.

1. Flip the binary representation's bits, turning 0s into 1s and 1s into 0s.
2. Increase the inverted value by 1.
3. The outcome is the negative number's two's complement.

By following these steps, you can find the two's complement representation of a given binary number, allowing for efficient representation and arithmetic operations on signed integers in digital systems.

Two's Complement Properties

Two's complement possesses several properties that make it a widely used and efficient method for representing signed integers in digital systems. Here are the key properties of the two's complement:

• Range of Representation:

 In an n-bit two's complement representation, the range extends from -2^(n-1) to 2^(n-1)-1. For example, in an 8-bit representation, the range is from -128 to 127. This range covers an equal number of positive and negative values, making it convenient for representing a wide range of numbers.

• Symmetric Range:

The range of representation in two's complement is symmetric, meaning there is an equal number of positive and negative values. This symmetry simplifies arithmetic operations as there is no need for separate handling of positive and negative numbers.

• Single Zero Representation:

Two's complement represents zero as all 0s, regardless of the sign bit. This eliminates the need for separate positive and negative zero representations, reducing complexity in calculations and allowing for a unique representation of zero.

• Addition and Subtraction:

The addition and subtraction of two's complement numbers can be performed using the same set of hardware and operations as those used for unsigned numbers. This property simplifies circuit design and allowsfor efficient arithmetic computations without requiring special handling for signed numbers.

• Overflow Detection:

 Overflow occurs when the result of an arithmetic operation exceeds the range of representation. In two's complement, overflow can be easily detected by comparing the carry-out of the sign bit with the carry-in. An overflow indicates that the result is invalid and requires additional handling to ensure accurate calculations.

• Efficient Hardware Implementation:

 Two's complement operations can be implemented using simple logic gates and circuits, making it efficient in terms of hardware requirements. This simplicity and efficiency have contributed to its widespread usage in processors, microcontrollers, and other digital systems.

• Compatibility with Unsigned Numbers:

Two's complement representation allows for seamless integration and compatibility with unsigned numbers. The same arithmetic operations and circuitry can be used for both signed and unsigned numbers, simplifying the design and implementation of digital systems.

Understanding these properties of two's complement is crucial for performing accurate arithmetic operations and handling signed integers efficiently in digital systems.

How to find the two’s complement of binary and decimal numbers?

Example 1

Determine two’s complement of (60)₁₀

Solution:

To find the two's complement of the decimal number (60)₁₀, follow these step-by-step instructions:

1. We first translate the decimal integer into binary form. In this case, the binary representation of (60)₁₀ is (00111100)₂.
2. Determine the number of bits you will use for the two's complement representation. Let's assume we're using an 8-bit representation for this example.
3. Verify the decimal number's sign. Since 60 is positive, the leftmost bit in the two's complement representation will be 0.
4. If the leftmost bit is 0 (indicating a positive number), the two's complement representation is the same as the binary representation. So, in this case, the two's complement of (60)₁₀ is (00111100)₂.

Example 2:

Calculate two’s complement if the decimal value is 5 and the size is 4 bits.

Solution
Number in decimal = 5

First, we have to convert 5 to binary:

Binary of 5 = 101

Selected Bits = 04

Binary Number after completing bits = 0101

Step 1:

Taking One's complement of binary number:

Write down the binary Number

 

Invert all values (Swap each 0 with 1 and each 1 with 0):

Step 2:

Taking Two's complement by adding 1 in the previous binary number:

 

Number in 2's complement with 04-bit representation

Decimal	5
Binary	0101
2's Complement	1011

A 2 complement calculator is an easy to use tool to calculate the two’s complement of positive and negative decimal number in a few seconds.

FAQs

Question 1:

 What is the purpose of using two's complement?

Answer:

 The two's complement representation is used to efficiently represent and perform arithmetic operations on signed integers in digital systems. It allows for a simple and unified approach to handle both positive and negative numbers using the same hardware and operations.

Question 2:

 How do I perform addition and subtraction using two's complement?

Answer:

 Addition and subtraction of numbers in two's complement are performed using the same hardware and operations as those used for unsigned numbers. To add or subtract two numbers, simply perform binary addition or subtraction, disregarding the sign bit. If an overflow occurs, it indicates an invalid result.

Question 3:

 Are two's complement representations standardized across different computer systems?

Answer:

 The concept of two's complement representation is standardized and widely used in digital systems. However, the number of bits used and specific implementation details may vary across different computer systems or programming languages. It's important to consider the specific system's bit width and representation conventions when working with two's complements.

Summary

In this article, we have discussed the definition of Two’s complement, solving methods, and properties of two’s complement. Also, the two’s complement will explain with the help of examples. After completely understanding this Article, anyone can defend easily this topic.  

Introduction

This article revolves around an integral aspect of academia: academic writing. This form of writing, distinct from casual communication, forms a crucial part of university and scholarly environments. 

Understanding the Task

Academic writing revolves around clarity, focus, and substantiated arguments. Before beginning, the first step is to comprehend the task at hand thoroughly. Read the assignment or essay question, paying attention to key instructions and terms. 

This knowledge will guide your writing process, whether you're asked to analyze, discuss, compare, or evaluate.

Before beginning your writing journey, you must:

- Carefully read your assignment or essay question to understand its requirements.

- Highlight key instructions and terms to guide your writing.

- Ascertain the type of response expected (analyze, discuss, compare, or evaluate).
- Use online tools such as Assignment Calculator for breaking down the task into manageable chunks.

The Importance of Planning

Moving on, we emphasize planning. Rather than diving into writing, take time to structure your thoughts. Develop a basic outline, consider your main points, and their interconnectedness. 

This skeleton will provide structure to your academic work, ensuring a smooth writing process later.

Planning forms the bedrock of structured academic writing. Here's how you can make the most of it:

- Take time to structure your thoughts before diving into writing.
- Develop a basic outline to provide structure to your work.
- Jot down main points and think about how they interrelate.
- Use online tools like Mindmeister to visually organise your ideas.

Writing Process

Entering the writing process might be daunting, but we'll break it down. Start with an introduction that outlines your topic and the direction your paper will take. Each subsequent paragraph should begin with a topic sentence introducing the main idea, followed by supporting arguments, and a summarising sentence linking back to your thesis.

Maintain clarity in your writing. Stay away from slang, jargon, and overly complex words. Your primary goal is effective communication, not impressing the reader with intricate language.

The writing process need not be daunting. Here's how you can approach it:

- Start with an introduction that outlines your topic and the direction your paper will take.
- Begin each subsequent paragraph with a topic sentence introducing the main idea, followed by supporting arguments, and a summarising sentence linking back to your thesis.
- Strive for clarity in your writing. Avoid slang, jargon, and overly complex words.
- Consider using paraphrasing tools like Paraphrasingtool.ai to rewrite sentences without losing their meaning, which is especially useful when summarising literature or avoiding plagiarism.

Conducting Research

Research forms an essential part of academic writing. Each point you make should have backing evidence from credible sources. Always note the source of your information, as accurate citation is crucial to avoid plagiarism.

Research underpins academic writing. To conduct it effectively, remember to:

- Substantiate each point you make with evidence from credible sources.
- Note the source of your information for proper citation, which is vital to avoid plagiarism.
- Use online databases like Google Scholar or JSTOR to access a wide range of scholarly articles and sources.

Proofreading and Revision

Proofreading should not be overlooked! Review your work for errors or ambiguity. Check the flow of your arguments and the organization of your paragraphs.

Proofreading and revision can greatly enhance your work:

- Thoroughly review your work for errors or ambiguity.
- Check for logical flow and organization in your paragraphs.
- Use online tools like Grammarly to catch grammatical errors and enhance readability.

Conclusion

Though academic writing might initially seem challenging, with practice, understanding, and patience, you'll gradually master it. 

Embrace the process: understand your task, plan your work, write clearly, and underpin your points with sound research. Always remember to proofread and revise meticulously. 

By following these steps, you'll develop into a proficient academic writer.

Do you want to improve your descriptive writing skills? Writing descriptively can take your readers on a journey through your stories and ideas. By mastering the art of descriptive writing, you can captivate your audience with vivid and powerful imagery.

In this article, we'll discuss three ways to improve your descriptive writing: using descriptive words and phrases, separating elements, and utilizing visuals.

With these tips, you can start crafting more evocative and engaging stories!

Use Descriptive Words And Phrases

Descriptive words and phrases are an essential tool for any writer looking to bring their work to life. They are used to help provide the reader with a vivid description of the scene, character, or emotion they are writing about, making the writing more engaging and interesting.

Descriptive words and phrases are important to use in descriptive writing as they can help to create a more vivid and immersive experience for the reader. Without them, the writing can feel flat and unengaging.

For example, if a writer were describing a character, they could use descriptive words such as “taciturn” and “wizened” to give the reader a greater sense of who the character is. They could also use phrases such as “his face was creased with age and experience” to further illustrate the character.

Using descriptive words and phrases in descriptive writing is not difficult, but some tips can help.

• Firstly, it is important to be precise with the language you use.
• Choose words and phrases that are specific and give the reader a clear sense of the scene or character.
• Avoid using clichés and overly-used phrases.
• Lastly, use sensory words and phrases to give the reader a greater sense of the scene.

Separate The Elements

Descriptive writing is a form of writing that focuses on creating a vivid picture of a person, place, or thing. It is used to describe and explain the details of a place, event, or character in a way that makes the reader feel like they are experiencing it themselves.The elements of descriptive writing are the five senses, strong verbs, figurative language, imagery, and precision.

Five Senses

The five senses are essential for descriptive writing because they provide readers with a vivid understanding of what’s happening. Examples of this could be describing the sound of laughter echoing through the room, the smell of freshly baked cookies, or the warmth of the sun on your face.

Strong Verbs

Strong verbs are also important for descriptive writing. Using more powerful, vivid verbs instead of the more basic ones can help create a stronger image in the reader's mind. An example of this might be “He lumbered across the room” instead of “He walked across the room.”

If you can’t figure out a strong word to use in lieu of a longer phrase, you can try taking online rephrasing. If you opt to rephrase online, you can learn a lot of descriptive words to use in your writing.

Figurative Languages& Imagery

Figurative languages like similes, metaphors, and personification can help to create a more vivid picture in the reader's mind.

Imagery can also be used, such as describing the colors, shapes, and textures of an object. As the name suggests, the purpose of imagery is to create an “image” in the mind of the reader.

Brevity

Although descriptive writing is inherently somewhat long-drawn, you have to be careful not to make it too lengthy. Considering that people can get somewhat bored with repetitive content, you should try and keep your writing concise.

If you can’t manage to keep your writing short, you can just write the full version and then use a summary generator afterward.

Utilize the Visuals

Visuals are an important part of any writing, especially in descriptive essays. Visuals can be anything from a picture or photograph to a diagram or illustration. They help to add a visual element to the writing, making it more interesting and easier to understand.

Visuals can be used to break up long, dense paragraphs and to make information easier to digest. For example, a diagram can provide a simple overview of a complex idea, or a picture can add color and life to an otherwise boring topic.

To use visuals for improving descriptive writing, it is important to ensure that the visuals are relevant to the topic and that they provide the necessary information. For example, if a text is describing the anatomy of a tree, a picture of a tree should show all of its components, such as the leaves, branches, trunk, and roots.

Visuals should be labeled and captioned appropriately so that readers can easily identify and understand them.When using visuals, it is important to remember to use them sparingly. Too many visuals can be distracting and can detract from the overall quality of the writing.

Final Words

Descriptive writing can be a powerful tool to communicate with readers and evoke emotion. By using descriptive words and phrases, separating elements, and utilizing visuals, writers can create vivid and engaging pieces of writing. With practice, any writer can improve their descriptive writing skills, and with a little bit of creativity and imagination, any writer can make their writing come alive.

Engaging content is the dream of every writer. So, which are the best tools to do it?

A writer in today's world has many responsibilities. Whether you're a student or a professional, you know you need to tend to a lot of guidelines, etc. So, having an assistant in the shape of a tool can help make it all easier for you.

But, finding a good assisting tool is difficult, considering there are many lackluster or bad tools on the internet. Therefore, let's talk about the best tools writers can use today.

Defining Engaging Content

A defining trait of engaging content isthat it is helpful, informative, and actionable. It's what you want to read because it provides you with something that you need.It's not just about the words on the screen but also about the page's design and how easy it is to navigate.

Engaging content is also interactive in a way where readers can see their own progress and get feedback from others. In order to have engaging content, it needs to be:

• helpful;
• effective;
• knowledgeable;
• informative and actionable.

On top of that, originality is also a key factor in making the content engaging.Plagiarism-free content is crucial for the success of your business.It will not only protect your business but will also make you look more trustworthy in the eyes of your customers.

3 Best Online Tools For Writers To Make Your Content Engaging

The best tools are the ones that assist in each key factor of writing. This means not only helping with grammatical issues or spelling but also with content tones and originality.

This means plagiarism checkers, paraphrasing tools, etc. Today, we're going to explore a blend of such tools. So, let's get started.

1. Plagiarism detector

Plagiarism is an unethical and dishonest act in its own right, but it also has negative effects on your content engagement.

The human brain is naturally good at recognizing patterns and things. So, people are able to recognize if they have read something before or not. So, if your content has a considerable amount of plagiarism in it, then it will be detected by your readers.

Now, you should know, people are always looking for new and stimulating things. It is hard for them to enjoy something they have already seen before. That is why by using plagiarized content, you make the engagement of your content go down. People will simply go away and look for something newer and fresher instead of reading something they probably already read before.

And for that reason, you should always utilize plagiarism-checking tools to make sure that none of your content is copied from anywhere. When you have made sure that all your content is unique, then you can publish it. Your readers will also appreciate the “uniqueness” aspect and engage with your content more.

2. Paraphrasing Tool

Paraphrasing tools are really useful for making your content unique and readable. They usually utilize NLP (Natural Language Processing)  and AI (Artificial Intelligence) to understand the given text and then make appropriate changes to it.

The best paraphrasing tools usually come with multiple modes. Some modes are only good for replacing words with their synonyms. Then other modes can change phrases as well as words. And some freemium tools have paid modes that can even change the sentence structure as well.

Usually, a mode for improving readability is also given in most paraphrasing tools. These modes basically change difficult words and phrases with their easier alternatives, thereby reducing the difficulty of understanding and reading the text.

If you don’t make your text readable, then its engagement suffers. Because people are unable to read it properly, they just leave instead.

Naturally, if something is easy to read, then people will read it till the end (provided that the content is interesting too). And that’s how engagement is improved with paraphrasing tools.

 

3. Content Analysis Tool

Content analyzers are quality tools that allow writers to make their content more engaging. Instead of delving into technical stuff, these tools keep it simple by allowing you to ensure that your content is readable.

Therefore, they urge you to use the right adverbs. And it focuses on using shorter sentences—which is one of the keys to engaging content. This makes it another viable addition to any writer's arsenal today. They can also advise you about passive voice usage and if there is too much of it in your content.

By acting on the advice provided by content analyzers, you can improve the engagement of your content.

Conclusion

These are the three best online tools writers can use today to make their content more engaging. It's important to use these tools and ensure that you employ them to So, use one or the other according to the need of your content at the time.

Every year we celebrate earth day on 22nd april and this festival is based on earth condition climatic condition global warming today pollution is a very danger enemy of ours pollution effects on global warming and pollution also effect the ozone layer it is also a very dangor problem pollyethene is a big cause of pollution trees are cutted down ver fast and the population is also a very big cause of pollution so let`s do a promise than crop one tree in in a monthand our duty never ends there we take care of plants and dont use plastic in huge amount happy earths day.

Education: Earth Day education programs provide educators, students, and the general public with resources and solutions to create a healthier, more sustainable planet. On Earth Day 20xx, our school instance, teachers and students in India participated in a day-long conference. At the conference, they learned about tree planting and care, participated in nature hikes, and presented their environmental action projects to the community.
Climate Change: Supporters raise awareness about climate change, human contribution to those changes, and opportunities to slow the phenomenon. We are facing geat sign of climate changes these which cause a large disturbance and worst damage on the earth so we need this improvemnet to protect our mother Earth and it's beauty.

अनुप्रयोग (APPLICATION)से बदलती दुनिया आज विश्व में अनुप्रयोगों की बहुत मांग है , अनुप्रयोग एक ऐसी चीज है जो आज के युग में बहुत आवश्यक है | इसको सिर्फ आधुनिक युग के नाना प्रकार के यन्त्र द्वारा ही चलाया जा सकता है | वह आधुनिक यन्त्र है स्मार्ट फ़ोन जिसको साधारणतया लोग "एंड्रॉइड" भी कहता है , एंड्राइड का मतलब होता है मनुष्ये के समान | इसके द्वारा हम अपने अनुप्रयोगों को चला सकते है |आज हम अपने ट्रेन, प्लैन आदि के टिकट अनुप्रयोग के द्वारा घर बेठे नाम दर्ज करवा सकते है | जैसा की हम जानते है की इसको चलाने के लिए एक यन्त्र का प्रयोग होता है जिसे हम एंड्रॉइड कहते है | इस एंड्रॉइड की शुरुआत "गूगल" द्वारा हुई थी |

आज गूगल ही अनुप्रयोग (application) बनाने का साधन नहीं है अपितु "फेसबुक" जैसी कम्पनी भी आज अनुप्रयोग बनाने में साकार हो गई है | आज के युग में हम कही गुम नहीं हो सकते क्योंकि "दिशा" बताने वाला अनुप्रयोग भी आज युग में उपस्थित है , जिसको "दिशा सूचक "कहते है| इसके द्वारा हम कही भी कभी भी अपनी दिशा का पता लगा सकते है | बाशर्ते की आपको इसे उपयोग करना आना चाहिए | यदि आपने इन्हें उपयोग करना सीख लिया तो समझ लो की आपने इस युग में चलना सीख लिया | आज कई ऐसे काम है जो हम अनुप्रयोग द्वारा मुशीबत को दूर कर सकते है जैसे की हमारी गाडी का पेट्रोल खत्म हो गया हो तो हम इंटरनेट द्वारा जहाँ हम खड़े है वहां का पता लगाएंगे और फिर वहां के "पेट्रोलपम्प" पर कॉल कर के पेट्रोल मंगवा सकते है |

अनुप्रयोग द्वारा हम घर बेठे सामान खरीद भी अपितु बेच भी सकते है | आज के युग में अनुप्रयोग पढ़ाई का एक मुख्य साधन बन चूका है जो की "एडोब रीडर" के नाम से मशहूर हुआ जो हमारी इंटरनेट द्वारा लाई गई किताबो को हमारे सामने प्रस्तुत करता है |अनुप्रयोग एक खेल के रूप में उभरा जो की मनोरंजन का साधन बना , अगर हम कही सफ़र में हो तो हम उदास नहीं होंगे सिर्फ हमें अपने स्मार्ट फ़ोन को उठाना है और सिर्फ एक क्लिक करना है | यह चीज छोटी सी है अपितु है बड़ी काम की | आप भी पता लगाइए कि आपके लिए कौन सा application उपयुक्त  है | 

 

 

कम्प्यूटर कम्प्यूटर एक प्रभावशाली मशीन है| यह विज्ञानं की सबसे अद्भुत खोज है| इसके द्वारा घंटो का काम चन्द मिनटों में निपटा लेते है| वैसे तो कम्प्यूटर का अविष्कार पहले एक कैलकुलेटर रूप में हुआ था ताकि जोड़ गुणा भाग जल्दी –जल्दी कर सके लेकिन इसकी उपयोगिता के कारण इसको और विकशित किया गया| कम्प्यूटर एक बिजली से चलने वाला यन्त्र है जिसके द्वारा हम कठिन से कठिन कार्य भी कम समय में कर सकते है | इसका अविष्कार सन 1823 मई चार्ल्स बेबेज नामक वैज्ञानिक ने किया था जो कि यह दुनिया की सबसे यह सबसे अच्छी चीज थी इसलिए उनको आधुनिक कम्प्यूटर विज्ञान का परम पिता कहा गया | कम्प्यूटर आज हमारे दैनिक जीवन में महत्वपूर्ण भूमिका निभा रहा है| जीवन के सभी मुख्य क्षेत्रो में कम्प्यूटर का बेहद इस्तेमाल शुरू हुआ है | जैसे: नाभिकीय भौतिक विज्ञान अनुसंधान, वास्तुशिल्प, ड्राफ्टिंग, प्रायोगिक परिणाम, तथा गणितीय समीकरण तथा उनका विशलेषण आदि| आज के युग में तो शोपिंग, रेल टिकट आरक्षण, होटल बुकिंग, नेट बैंकिंग जैसी सभी चीजे कम्प्यूटर से किया जाता है| इसके द्वारा हम घर बैठे सारी दुनिया का हाल ले सकते है| जहाँ कम्प्यूटर ने अपने अच्छे प्रभाव दिखाए वही उनके बहुत से दुष्प्रभाव भी है जैसे बेरोजगारी (क्यों की यह कई लोगो का काम अकेले ही कर देता है),यह बच्चों पर भी बुरा प्रभाव डालता है, मानव को छोटे से क्षेत्र में रहने के लिए मजबूर कर दिया, इससे बेकारी बढती है आदि | अतः हमें ज्यादा से ज्यादा ज्ञान बढ़ाना चाहिए न की इसके चक्कर में फंसना चाहिए | जिस प्रकार बगुला दूध पीने के बाद पानी निकलता है उसी प्रकार हमें भी कम्प्यूटर से अच्छे काम करने चाहिए न की बुरे | 

A few heartily who love the creatures they feed them, provide shelters and pet them as a family members. This love to creature is viewable somewhere. They provide them all medical facilities and nutritional food, but some show cruelty towards these speechless creatures. They deal with them in very badly manner. I think this is a crime against them. They also have right to live and get love and sympathy. After getting love and sympathy animals feel them.   

A few heartily who love the creatures they feed them, provide shelters and pet them as a family members. This love to creature is viewable somewhere. They provide them all medical facilities and nutritional food, but some show cruelty towards these speechless creatures. They deal in very badly manner. I think this is a crime against them. They also have right to live and get love and sympathy. After getting love and sympathy animals feel them.   

 

यह एक बहुत अच्छा  विषय है ! हर विद्यार्थी की यदि ईच्छा होती है की वह हर गणित के सवाल को अच्छी तरह हल कर सके | और लगभग सभी विद्यार्थी की पहली पसंद की विषय निसंदेह गणित (Mathematics) ही है | 

This is a very intresting topic "How to obtain good marks in exams" for all students. Hard working is another thing, but every desire can not be aquired by hard working. Wisdom is another thing that makes possible every thing in this world. Understanding is an another thing which creates a creativity that gives way to go ahead. There are some points which can be done in special manner to obtain good marks. These are as follows.

• Understanding topics and terms.
• Practicing topics and terms in that way by which you have understood.
• Try to remind them after 2 to 3 days of interval.
• Build your memory.

Understanding topics and terms of a subject is very important otherwiase it can not be learnt in right manner, whenever you can forget in exams. Because you have only learnt not understood. 

हम सभी गर्मी की छुट्टिय मनाने कही न कही जाते है परन्तु मसूरी में जाकर छुट्टिया मनाने की बात ही कुछ और है | 

आप सभी एक बार वहां अवश्य जाये | मैं अब तक वहां नहीं गया हूँ परन्तु लोगों से अवश्य ही सुना हु कि वह एक सुन्दर स्थान है | यहाँ मौसम हमेशा सुहावना रहता है |