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## Introduction

### What is integral calculus?

Integral calculus is a branch of mathematics that deals with finding the total amount or area of something by adding up infinitely small parts. For example, if you want to find the area under a curve, you can divide it into thin rectangles and add up their areas. However, this method is not very accurate, because there will be some gaps or overlaps between the rectangles. To get a more precise answer, you need to use integral calculus, which allows you to find the exact area by using a special symbol called an integral. An integral represents the sum of infinitely many infinitesimal parts, which are so small that they fit perfectly under the curve. The process of finding an integral is called integration.

Gorakh Prasad was a renowned Indian mathematician and educator who lived from 1876 to 1966. He was a professor of mathematics at Queen's College in Varanasi for over 40 years, and he wrote several books on various topics in mathematics, such as algebra, trigonometry, geometry, differential calculus, and integral calculus. His books were widely used in India and abroad as textbooks for undergraduate and graduate students. He was also known for his contributions to mathematical research, especially in number theory and analysis. He received many honors and awards for his work, such as the Padma Shri in 1957 and the Srinivasa Ramanujan Medal in 1961.

### Why should you read his book?

There are many reasons why you should read Integral Calculus by Gorakh Prasad if you want to learn or improve your skills in integral calculus. Here are some of them:

• The book covers all the topics that are essential for understanding integral calculus, such as indefinite integrals, definite integrals, methods of integration, improper integrals, applications of integration in geometry, physics, engineering, etc.

• The book explains each concept and technique in a clear and concise manner, with examples and illustrations that make it easy to follow and comprehend.

• The book provides numerous solved examples and exercises that help you practice and test your knowledge and skills. The solutions are given at the end of each chapter for your reference.

• The book shows you how to apply integral calculus in various fields and situations, such as finding areas, volumes, lengths, centroids, moments of inertia, work done by forces, etc.

• The book is suitable for both beginners and advanced learners, as it starts from the basics and gradually progresses to more complex and challenging topics.

## Main features of the book

### Clear and concise explanations

One of the main features of Integral Calculus by Gorakh Prasad is that it gives clear and concise explanations of each concept and technique in integral calculus. The author uses simple language and avoids unnecessary jargon and technicalities. He also uses diagrams and graphs to illustrate the ideas and methods. For example, here is how he explains the concept of an indefinite integral:

An indefinite integral of a function f(x) is a function F(x) whose derivative is f(x). That is, F'(x) = f(x). The symbol f(x)dx denotes an indefinite integral of f(x), and it is read as "the integral of f(x) with respect to x". The function F(x) is called an antiderivative or a primitive of f(x), and it represents the general solution of the equation F'(x) = f(x). Since the derivative of a constant is zero, adding any constant to F(x) will not change its derivative. Therefore, there are infinitely many antiderivatives of f(x), and they differ only by a constant. This constant is called the constant of integration, and it is usually denoted by C. Thus, we write f(x)dx = F(x) + C, where C is an arbitrary constant.

### Numerous solved examples and exercises

Another main feature of Integral Calculus by Gorakh Prasad is that it provides numerous solved examples and exercises that help you practice and test your knowledge and skills in integral calculus. The solved examples are given throughout each chapter, and they show you how to apply the concepts and techniques to solve various problems. The exercises are given at the end of each chapter, and they range from simple to difficult, covering all the topics in the chapter. The solutions are given at the end of the book for your reference. For example, here is one of the solved examples from the chapter on methods of integration:

Example: Find (x^2 + 1)/(x^4 + 1)dx.

Solution: We use the method of partial fractions to write the integrand as a sum of simpler fractions. We have

(x^2 + 1)/(x^4 + 1) = (Ax + B)/(x^2 + x + 1) + (Cx + D)/(x^2 - x + 1)

Multiplying both sides by x^4 + 1, we get

x^2 + 1 = (Ax + B)(x^2 - x + 1) + (Cx + D)(x^2 + x + 1)

Comparing the coefficients of x^3, x^2, x, and the constant term on both sides, we get

A + C = 0

B - A - C - D = 0

-B - D = 0

B + D = 1

Solving these equations, we get A = -1/4, B = 1/2, C = 1/4, D = 1/2. Therefore,

(x^2 + 1)/(x^4 + 1) = (-x/4 + 1/2)/(x^2 + x + 1) + (x/4 + 1/2)/(x^2 - x + 1)

Now we can integrate each fraction separately using the substitution method. We have

(-x/4 + 1/2)/(x^2 + x + 1)dx = -(u/4 - 1/2)/u^2 du, where u = x^2 + x + 1

= -(lnu/4 - u^-1/2) + C_1

= -(lnx^2 + x + 1/4 - (x^2 + x + 1)^-1/2) + C_1

Similarly,

(x/4 + 1/2)/(x^2 - x + 1)dx = (u/4 - 1/2)/u^2 du, where u = x^2 - x + 1

= (lnu/4 - u^-1/2) + C_2

= (lnx^2 - x + 1/4 - (x^2 - x + 1)^-1/2) + C_2

Adding these two integrals, we get

(x^2 + 1)/(x^4 + 1)dx = (lnx^2 - x + 1 - lnx^2 + x + 1)/4 - ((x^2 - x + 1)^-1/2 - (x^2 + x + 1)^-1/2)/2 + C

where C = C_1 + C_2 is an arbitrary constant.

### Applications of integral calculus in various fields

Another main feature of Integral Calculus by Gorakh Prasad is that it shows you how to apply integral calculus in various fields and situations. The book has a separate chapter on applications of integration, where it explains how to use integrals to find areas, volumes, lengths, centroids, moments of inertia, work done by forces, etc. The book also gives examples of how integral calculus is used in physics, engineering, economics, biology, etc. For example, here is how the book explains how to use integrals to find the area of a circle:

Example: Find the area of a circle with radius r.

Solution: We can use the method of slicing to divide the circle into thin sectors and add up their areas. However, this method is not very convenient, because we need to use trigonometry to find the area of each sector. A better way is to use the method of cylindrical shells. We can imagine that the circle is formed by rotating a line segment of length r around the x-axis. Then we can divide the line segment into thin strips and rotate each strip around the x-axis to form a cylindrical shell. The area of each shell is equal to the circumference of the base times the height. The circumference of the base is 2πy, where y is the distance from the x-axis to the strip. The height is dx, where x is the distance from the y-axis to the strip. Therefore, the area of each shell is 2πydx. To find the total area of the circle, we need to add up the areas of all the shells from x = 0 to x = r. This means we need to evaluate the integral 0r 2πydx. To do this, we need to express y in terms of x. Since the line segment lies on the circle with equation x^2 + y^2 = r^2, we can solve for y and get y = (r^2 - x^2). Substituting this into the integral, we get

0r 2πydx = 0r 2π(r^2 - x^2)dx

We can use a trigonometric substitution to evaluate this integral. We let x = r sin θ and dx = r cos θ dθ. Then we have

0r 2π(r^2 - x^2)dx = 0π/22π(r^2 - r^2 sin^2 θ)r cos θ dθ

= 0π/22πr(1 - sin^2 θ) cos θ dθ

= 0π/22πr cos^2 θ dθ

= πr0π/22 cos^2 θ dθ

We can use a trigonometric identity to simplify this integral. We have cos^2 θ = (1 + cos 2θ)/2. Therefore,

0π/22 cos^2 θ dθ = 0π/22 (1 + cos 2θ)/2 dθ

= (θ/2 + sin 2θ/4)0π/2

= (π/4 + sin π/4)/4 - (0 + sin 0)/4

= π/16 + (2)/16 - 0

= π/16 + (2)/16

Multiplying this by πr, we get

0r 2πydx = πr(π/16 + (2)/16)

= π^2r/16 + π(2)r/16

This is the area of the circle with radius r.

• You can access the book anytime and anywhere, as long as you have a device that can read PDF files.

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• Search for Integral Calculus by Gorakh Prasad in the search bar. You can also use the ISBN number of the book, which is 9788171940091.

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While downloading Integral Calculus by Gorakh Prasad in PDF format online can be convenient and beneficial, you also need to take some precautions to avoid any problems or risks. Here are some of them:

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## Conclusion

### Summary of the main points

In this article, we have given you an overview of what integral calculus is, who Gorakh Prasad is, why you should read his book, what are the main features of the book, how to download the book in PDF format, and what are some FAQs related to the topic. We hope that this article has helped you gain a better understanding of integral calculus and how to learn it from Gorakh Prasad's book. Here are some of the main points we have covered:

• Integral calculus is a branch of mathematics that deals with finding the total amount or area of something by adding up infinitely small parts.

• Gorakh Prasad was a renowned Indian mathematician and educator who wrote several books on various topics in mathematics, including integral calculus.

• Integral Calculus by Gorakh Prasad is one of the most popular and widely used textbooks on integral calculus, and it has been praised by students and teachers alike for its clarity, simplicity, and comprehensiveness.

• The book explains each concept and technique in a clear and concise manner, with examples and illustrations that make it easy to follow and comprehend.

• The book provides numerous solved examples and exercises that help you practice and test your knowledge and skills.

• The book shows you how to apply integral calculus in various fields and situations, such as finding areas, volumes, lengths, centroids, moments of inertia, work done by forces, etc.

• You can download the book in PDF format online from a reliable website that offers free or paid downloads of mathematical books.

• You can enjoy the benefits of downloading the book online, such as accessing it anytime and anywhere, saving money and space, searching for keywords and phrases, zooming in and out of the pages, printing out the pages you need, etc.

• You need to take some precautions while downloading the book online, such as having a stable and secure internet connection, having enough storage space on your device, having a good antivirus software on your device, downloading the book from a reputable and legal website, respecting the author's rights and intellectual property, etc.

## FAQs

### Q: What is the difference between indefinite and definite integrals?

A: An indefinite integral of a function f(x) is a function F(x) whose derivative is f(x). An indefinite integral represents the general solution of the equation F'(x) = f(x), and it has an arbitrary constant C. A definite integral of a function f(x) is a number that represents the area under the curve of f(x) between two limits a and b. A definite integral is calculated by finding an antiderivative F(x) of f(x) and evaluating it at the limits a and b. A definite integral is written as ab f(x)dx = F(b) - F(a).

### Q: What are some methods of integration?

A: Some methods of integration are:

• Substitution method: This method involves replacing a part of the integrand with another variable u that is easier to integrate. The substitution must be done for both the function and the differential.

• Integration by parts: This method involves using the product rule of differentiation in reverse. It allows us to integrate products of functions by splitting them into two parts u and v.

• Partial fractions method: This method involves decomposing a rational function into simpler fractions that are easier to integrate. The decomposition must be done according to the degree and type of the denominator.

• Trigonometric substitution method: This method involves replacing a part of the integrand with a trigonometric function that simplifies the expression. The substitution must be done using appropriate identities and triangles.

### Q: What are some applications of integration?

A: Some applications of integration are:

• Finding areas: Integration can be used to find the area under a curve, between two curves, or bounded by a closed curve.

• Finding volumes: Integration can be used to find the volume of a solid that is obtained by rotating a curve around an axis or by slicing a solid into thin cross-sections.

• Finding lengths: Integration can be used to find the length of a curve or an arc of a curve.

• Finding centroids: Integration can be used to find the center of mass or the balance point of a plane region or a solid.

• Finding moments of inertia: Integration can be used to find the resistance of a plane region or a solid to rotation around an axis.

• Finding work done by forces: Integration can be used to find the work done by a variable force that acts along a curve or a surface.

### Q: How can I check my answers when I do integration problems?

A: One way to check your answers when you do integration problems is to differentiate your answer and see if you get back the original integrand. For example, if you have x^2dx = x^3/3 + C, you can differentiate x^3/3 + C and get x^2, which is the original integrand. Another way to check your answers is to use an online calculator or software that can perform integration, such as Wolfram Alpha, Symbolab, Desmos, etc. You can compare your answer with the answer given by the calculator or software and see if they match.

### Q: Where can I find more resources on integral calculus?

A: There are many resources on integral calculus that you can find online or offline. Some examples are:

• Online courses: There are many online courses that teach integral calculus, such as Khan Academy, Coursera, edX, Udemy, etc. You can enroll in these courses and learn from videos, lectures, quizzes, assignments, etc.

• Online videos: There are many online videos that explain integral calculus, such as YouTube, Khan Academy, MathIsFun, etc. You can watch these videos and learn from examples, animations, demonstrations, etc.

• Online books: There are many online books that cover integral calculus, such as OpenStax, MIT OpenCourseWare, Project Gutenberg, etc. You can read these books and learn from definitions, theorems, proofs, exercises, etc.

Online forums: There are many online forums where you can ask questions and get answers on integral calculus, such as Reddit, Quora