Integral calculus: what is it and how to calculate it?

Integral calculus is a widely used concept in mathematics that inverses the process of differential. In mathematics, calculus has two main branches one is integral calculus and the other is differential calculus. 

The differential calculus is used to calculate the slope of the tangent line while the integral calculus is used to find the area under the curve. In this post, we will discuss integral calculus along with an explanation and calculation.

Integral calculus

A term used to determine the new function or the numerical value of the function whose original function is derivative is known as the integral calculus. The area under the curve can be calculated with the help of this type of calculus.

For determining the numerical result of the function, it takes the upper and lower limit value of the function i.e., the boundary values of the function. The integral is solved with the help of taking the integrating variable of the function. 

Laws of integral calculus 

The basics rules of integral are:

Types of integral calculus explained with examples

Below are two commonly used types of integral calculus.

• Definite integral
• Indefinite integral

These types of integral calculus can be found quite easily by using an integral calculator that provides steps to make it more understanding.

Let us discuss the types of integral calculus along with examples.

Definite integral 

In integral calculus, to find the numerical values of the function by taking the boundary values a definite integral is used. First of all, you have to integrate the function with respect to the integrating variable.

After that apply the boundary values of the function with the help of the fundamental theorem of calculus. This theorem states that the upper limit of the function must be applied first to the function then the lower limit with a negative sign among them. 

The general formula for the definite integral is:

ab m(x) dx = M(b) – M(a) = N

Where, a & b are the lower and the upper limit of the function respectively, m(x) is the integrand function, x is the variable of integration, M(b) – M(a) is the fundamental theorem of the calculus, and N is the final result of the function.

Example

Evaluate the definite integral of m(x) = 12x³ – 14x³ – 10x⁴ + 15cos(x) + 5 with respect to x & the [2, 3] is the interval.

Solution 

Step I: Take the given integrand and apply the integral notation to it.

m(x) = 12x³ – 14x³ – 10x⁴ + 15cos(x) + 5 

23 [m(x)] dx = 23 [12x³ – 14x³ – 10x⁴ + 15cos(x) + 5] dx

Step II: Use the sum and difference laws of integral calculus and apply the integral notation to each function separately.

23 [12x³ – 14x³ – 10x⁴ + 15cos(x) + 5] dx = 23 [12x³] dx – 23 [14x³] dx – 23 [10x⁴] dx + 23 [15cos(x)] dx + 23 [5] dx

Step III: Now use the constant function law of integral and take the constant coefficients outside the integral notation.

= 1223 [x³] dx – 1423 [x³] dx – 1023 [x⁴] dx + 1523 [cos(x)] dx + 23 [5] dx

Step IV: Use the power and trigonometry law of integral calculus and solve the above expression. 

= 12 [x³⁺¹ / 3 + 1]³₂ – 14 [x³⁺¹ / 3 + 1]³₂ – 10 [x⁴⁺¹ / 4 + 1]³₂ + 15 [sin(x)]³₂ + [5x]³₂ 

23 [12x³ – 14x³ – 10x⁴ + 15cos(x) + 5] dx = 12 [x⁴ / 4]³₂ – 14 [x⁴ / 4]³₂ – 10 [x⁵ / 5]³₂ + 15 [sin(x)]³₂ + [5x]³₂ 

= 12/4 [x⁴]³₂ – 14/4 [x⁴]³₂ – 10/5 [x⁵]³₂ + 15 [sin(x)]³₂ + 5 [x]³₂ 

= 3 [x⁴]³₂ – 7/2 [x⁴]³₂ – 2 [x⁵]³₂ + 15 [sin(x)]³₂ + 5 [x]³₂ 

Step V: Apply the boundary values.

= 3 [3⁴ – 2⁴] – 7/2 [3⁴ – 2⁴] – 2 [3⁵ – 2⁵] + 15 [sin(3) – sin(2)] + 5 [3 – 2] 

= 3 [81 – 16] – 7/2 [81 – 16] – 2 [243 – 64] + 15 [sin(3) – sin(2)] + 5 [3 – 2] 

= 3 [65] – 7/2 [65] – 2 [179] + 15 [sin(3) – sin(2)] + 5 [1] 

= 195 – 455/2 – 358 + 15 [sin(3) – sin(2)] + 5 

= 195 – 227.5 – 358 + 15 [sin(3) – sin(2)] + 5 

= –395.5 + 15sin(3) – 15sin(2) 

Indefinite integral 

In integral calculus, to find the new function whose original function is derivative an indefinite integral is used. It does not have any boundary values as limit values are used to find the numerical result not for finding the new function.

It integrates the function with respect to the integrating variable. The general formula for the indefinite integral is:

ʃ m(x) dx = M(x) + C

 Where m(x) is the integrand function, x is the variable of integration, M(x) is a new function, and C is the constant of integration. 

Example 

Evaluate the indefinite integral of m(x) = 15x⁴ – 20x + 6x³ – cos(x) + 21 with respect to x. 

Solution 

Step I: Take the given integrand and apply the integral notation to it.

m(x) = 15x⁴ – 20x + 6x³ – cos(x) + 21 

ʃ m(x) dx = ʃ [15x⁴ – 20x + 6x³ – cos(x) + 21] dx

Step II: Use the sum and difference laws of integral calculus and apply the integral notation to each function separately.

ʃ [15x⁴ – 20x + 6x³ – cos(x) + 21] dx = ʃ [15x⁴] dx – ʃ [20x] dx + ʃ [6x³] dx – ʃ [cos(x)] dx + ʃ [21] dx

Step III: Now use the constant function law of integral and take the constant coefficients outside the integral notation.

= 15ʃ [x⁴] dx – 20ʃ [x] dx + 6ʃ [x³] dx – ʃ [cos(x)] dx + ʃ [21] dx

Step IV: Use the power and trigonometry law of integral calculus and solve the above expression.  

= 15 [x⁴⁺¹ / 4 + 1] – 20 [x¹⁺¹ / 1 + 1] + 6 [x³⁺¹ / 3 + 1] – [sin(x)] + [21x]

= 15 [x⁵ / 5] – 20 [x² / 2] + 6 [x⁴ / 4] – [sin(x)] + [21x]

= 15/5 [x⁵] – 20/2 [x²] + 6/4 [x⁴] – [sin(x)] + [21x]

= 3 [x⁵] – 10 [x²] + 3/2 [x⁴] – [sin(x)] + [21x]

= 3x⁵ – 10x² + 1.5x⁴ – sin(x) + 21x

Use an antiderivative calculator to check the result of the above example. 

Conclusion 

Now the confusion about solving the complex problems of integral calculus must be cleared. As all the basics of integral calculus along with explanations and calculations are covered in this post. 

SSC CGL 2022 Notification

SSC CGL 2022 Notification: The Staff Selection Commission has released SSC CGL 2022 Notification on its official website. This time SSC has made some changes in the Exam pattern and syllabus of the SSC CGL examination 2022. In this article, you can find SSC CGL 2022 Notification with the changed Exam Pattern and New Syllabus for Tier-1 and Tier-2 Examination.

All details related to SSC CGL 2022 Notification can be found in this article. SSC CGL 2022 Eligibility, SSC CGL 2022 Vacancy, SSC CGL 2022 Important Dates, SSC CGL 2022 New Exam Pattern for Mains, SSC CGL 2022 New Syllabus, etc. can be seen in this article. All interested candidates must go through this article to get the complete details of the SSC CGL 2022 Notification.

SSC CGL 2022 Basic Information– Pattern Changed

It is a very good chance for all aspirants to get a very prestigious job from SSC CGL 2022 Exam. All interested candidates will have to have a clear idea of the exam. So, in the table below, you will find all the information related to SSC CGL 2022 Notification, Exam Pattern, and Selection Process. This exam is a graduate-level exam, which had 4 stages of selection, but this time SSC has changed the Pattern for the recruitment of the SSC CGL 2022 Exam. This time the Selection process for SSC CGL 2022 Exam has just 2 Stages Prelims and Mains, That means there are No Tier-3 and Tier-4 Exams this time in SSC CGL 2022 exam Notification.

SSC CGL 2022 Exam Date

SSC CGL 2022 Application date is 17^th September 2022 and SSC CGL last date to apply is 8th October 2022. The SSC CGL 2022 Tier-1 Exam will be conducted in December 2022. Candidates can check the dates of other events related to SSC CGL 2022 in the table below.

SSC CGL 2022 Vacancy

The SSC CGL 2022 Vacancy as per the official notification and more than 20,000+ this year. Last year The SSC CGL 2021 vacancy was just 7686. The trend for the vacancy in SSC CGL 2022 has been quite similar for the years 2018, 2019, 2020, and 2021. but it is apparently different this time, the tentative vacancies for SSC CGL 2022 are approximately 3 times as compared to the trend of the last three years.

The table below shows the SSC CGL 2022 category vacancy last year and the past few years. But in SSC CGL 2022 there are tentative vacancies of around 20,000+ due to the inclusion of posts from other examinations as well and many other factors are responsible for this massive increase in the number of vacancies for the SSC CGL 2022 Examination.

SSC CGL 2022 Eligibility Criteria

SSC CGL qualification is graduation from any stream from a recognized board. But there are some specifications according to the posts which are mentioned here. Candidates must check the SSC CGL Age limit before they apply for the exam.

SSC CGL age limit

The age limit for SSC CGL 2022 is 18-32. However, there are many posts that ask for a particular age group to apply. So, candidates must be very careful while applying for the posts.

SSC CGL qualification

SSC CGL qualification is graduation from any recognized institute in any stream. However, there are some posts that require the candidates to have some desirable qualifications. Here we have tried to solve all confusion regarding SSC CGL qualification.

Assistant Audit Officer/Assistant Accounts Officer:

Essential Qualifications: Bachelor’s Degree from a recognized University or Institute.

Desirable Qualifications: Chartered Accountant or Cost & Management Accountant or Company Secretary or Masters in Commerce or Masters in Business Studies or Masters in Business Administration (Finance) or Masters in Business Economics.

Junior Statistical Officer:

Bachelor’s Degree in any subject from a recognized University or Institute with at least 60% Marks in Mathematics at 12th standard level; Or Bachelor’s Degree in any subject with Statistics as one of the subjects at degree level.

Statistical Investigator Grade-II:

Bachelor’s Degree in any subject with Statistics as one of the subjects from a recognized University or Institute. The candidates must have studied Statistics as a subject in all the three years or all the 6 semesters of the graduation course.

For All Other Posts:

Bachelor’s Degree from a recognized University or equivalent in any subject or discipline.

SSC CGL 2022 Selection Process

There is some change in SSC CGL Selection Process 2022, it has now only two tiers i.e Tier-1 and Tier-2 computer-based examinations( from SSC CGL 2022 – the pattern has changed). SSC CGL Tier-1 is a computer-based test with multiple type questions. Each tier is designed to assess the ability of the candidate. SSC CGL Exam Pattern for each tier is different and is mentioned below.

SSC CGL Tier-1 2022 Exam Pattern Change:

This is an online test with four sections General Awareness, English Language, General Intelligence and Reasoning, and Quantitative Aptitude. 2 marks are awarded for each correct answer and a 0.25 mark is deducted for each wrong answer. The candidates are advised to carefully read the new Notification to know about the change in exam pattern and syllabus to perform well in the examination.

SSC CGL Tier-2 2022 Exam Pattern Change:

There is some change in SSC CGL Tier-2 2022 Exam Pattern. It will now have only three papers papers which are divided into different sections and modules. Paper- 1 will contain three sections comprising 5 modules( Quantitative Aptitude, English Comprehension, General Awareness, Reasoning, Computer knowledge). All the candidates who qualify the tier-1 are eligible to appear for the SSC CGL tier-2. Candidates with an Economics background can take the Finance and Economics paper. There will be seperate papers for the Posts of JSO/AAO. paper-II(Statistics) paper is for those who opt for JSO post, whereas Paper-III(Finance and Accounting) is for those who opt for Asst. Audit Officer.

• Tier II will include conducting Paper-I, Paper-II, and Paper-III in separate shift(s)/ day(s).
• Paper-I is compulsory for all the posts.
• Paper-II will be for only those candidates who apply for the posts of Junior Statistical Officer (JSO) in the Ministry of Statistics and Programme Implementation and who are shortlisted in Tier-I for these Posts.
• Paper-III will be for only those candidates who are shortlisted in Tier-I for Paper-III i.e. for the posts of Assistant Audit Officer/ Assistant
• Accounts Officer.
• Paper-I will include the following three sections having two modules each:
• Section-I :
• Module-I:Mathematical Abilities
• Module-II: Reasoning and General Intelligence.
• Section-II :
• Module-I: English Language and Comprehension
• Module-II: General Awareness
• Section-III :
• Module-I: Computer Knowledge Test
• Module-II: Data Entry Speed Test

SSC CGL Syllabus 2022-23 Tier-1 and Tier-2

SSC CGL Syllabus 2022-23 includes topics from four sections: Quantitative Aptitude, General Intelligence and Reasoning, English Language, and General Awareness. Here we have compiled all the important topics from all four sections.

Indicative Syllabus (Tier-I):
General Intelligence & Reasoning: It would include questions of both verbal and non-verbal types. This component may include questions on analogies, similarities and differences, space visualization, spatial orientation, problem-solving, analysis, judgment, decision making, visual memory, discrimination, observation, relationship concepts, arithmetical reasoning and figural classification, arithmetic number series, non-verbal series, coding and decoding, statement conclusion, syllogistic reasoning, etc. The topics are, Semantic Analogy, Symbolic/Number Analogy, Figural Analogy, Semantic Classification, Symbolic/Number Classification, Figural Classification, Semantic Series, Number Series, Figural Series, Problem Solving, Word Building, Coding & de-coding, Numerical Operations, symbolic Operations, Trends, Space Orientation, Space Visualization, Venn Diagrams, Drawing inferences, Punched hole/ pattern- folding& un-folding, Figural Pattern-folding and completion, Indexing, Address matching, Date & city matching, Classification of center codes/roll numbers, Small & Capital letters/numbers coding, decoding and classification, Embedded Figures, Critical thinking, Emotional Intelligence, Social Intelligence.

General Awareness: Questions in this component will be aimed at testing the candidates’ general awareness of the environment around him and its application to society. Questions will also be designed to test knowledge of current events and of such matters of every day observations and experience in their scientific aspect as may be expected of any educated person. The test will also include questions relating to India and its neighbouring countries especially pertaining History, Culture, Geography, Economic Scene, General Policy & Scientific Research.

Quantitative Aptitude: The questions will be designed to test the ability of appropriate use of numbers and number sense of the candidate. The scope of the test will be computation of whole numbers, decimals, fractions and relationships between numbers, Percentage. Ratio & Proportion, Square roots, Averages, Interest, Profit and Loss, Discount, Partnership Business, Mixture and Alligation, Time and distance, Time & Work, Basic algebraic identities of School Algebra & Elementary surds, Graphs of Linear Equations, Triangle and its various kinds of centres, Congruence and similarity of triangles, Circle and its chords, tangents, angles subtended by chords of a circle, common tangents to two or more circles, Triangle, Quadrilaterals, Regular Polygons, Circle, Right Prism, Right Circular Cone, Right Circular Cylinder, Sphere, Hemispheres, Rectangular Parallelepiped, Regular Right Pyramid with triangular or square base, Trigonometric ratio, Degree and Radian Measures, Standard Identities, Complementary angles, Heights and Distances, Histogram, Frequency polygon, Bar diagram & Pie chart.

English Comprehension: Candidates’ ability to understand correct English, their basic comprehension and writing ability, etc. would be tested.

SSC CGL 2022 Application Process

SSC CGL Application process includes several steps that need to be performed to successfully submit your application for the exam. SSC CGL exam fees is Rs. 100 for general/OBC Category males and SSC CGL application fees for females is Rs. 0. SSC CGL apply online on the official website.

• Applicants must visit the official website https://ssc.nic.in to fill the form.
• Then Fill the information asked in the form.
• Then you have to upload your photo.
• In the online Application Form, candidates are required to upload the scanned color passport size photograph in JPEG format (20 KB to 50 KB). The photograph should not be more than three months old from the date of publication of the Notice of Examination. Image dimension of the photograph should be about 3.5 cm (width) x 4.5 cm (height).
• The photograph should be without cap, spectacles and both ears should be visible.
• Once you submit the form, you will have to pay.
• Choose the payment choices if you want to pay online or through offline challan.
• Once you make the payment your application process is over.
  

Application Fees

Fee payable: ₹100/- (Rs. one hundred only).
Women candidates and candidates belonging to Scheduled Castes (SC), Scheduled Tribes (ST), Persons with Disabilities (PwD) and Ex-servicemen (ESM) eligible for reservation are exempted from payment of fee.

SSC CGL 2022 Notification out

SSC CGL 2022 Notification is out on the official website and candidates can check it. SSC CGL 2022 is an all-India exam and SSC CGL Eligibility criteria 2022 is any graduate from a government-recognized board. Apply for SSC CGL 2022 online on the official website before 08th October 2022. SSC CGL Application process, SSC CGL Syllabus 2022-23, and all the other details of the SSC CGL Notification are mentioned in this article. The last date to apply for SSC CGL 2022 Exam is 8 October 2022.

Some Important Links

FAQs