# The Benefits of Reading Integral Calculus by Hari Kishan: A Book that Explains Integration in a Clear and Simple Way

## Integral Calculus by Hari Kishan pdf: A Comprehensive Guide for Students

Integral calculus is one of the most important branches of mathematics that deals with finding the total amount or net change of a quantity that varies continuously. It has many applications in physics, engineering, economics, biology and other fields. For example, integral calculus can help us find the area under a curve, the length of a curve, the volume and surface area of solids of revolution, the work done by a force, the center of mass of a system, and so on.

## Integral Calculus By Hari Kishan pdf

If you are a student who wants to learn integral calculus or improve your skills in this subject, you might be looking for a good textbook that covers all the basic and advanced concepts in a clear and rigorous way. One such textbook is Integral Calculus by Hari Kishan, published by Atlantic Publishers & Dist in 2005. This book is a unique textbook on integration that aims at providing a fairly complete account of the basic concepts required to build a strong foundation for a student endeavouring to study this subject.

In this article, we will give you an overview of what this book offers and why you should read it. We will also provide you with some useful information on where to get the pdf version of the book and how to use it effectively. Let's get started!

## Basic Concepts of Integral Calculus

The first part of the book covers the basic concepts of integral calculus that you need to know before moving on to more advanced topics. These concepts include:

Antiderivatives and indefinite integrals: An antiderivative of a function f(x) is another function F(x) such that F'(x) = f(x) for all x in the domain of f(x). An indefinite integral of f(x) is the general form of all antiderivatives of f(x), denoted by f(x)dx = F(x) + C, where C is an arbitrary constant.

Methods of integration: There are various methods or techniques that can help us find indefinite integrals or antiderivatives of different types of functions. Some of these methods are:

### Integration by substitution

This method involves replacing a variable or expression in the integrand with another variable or expression that makes the integration easier. For example, if we want to find (2x + 1)^3 dx, we can substitute u = 2x + 1 and du = 2dx, and then integrate u^3 (du/2) = (u^4/8) + C = ((2x + 1)^4/8) + C.

### Integration by parts

This method involves using the product rule for differentiation in reverse. It states that u dv = uv v du, where u and v are functions of x. For example, if we want to find x cos x dx, we can choose u = x and dv = cos x dx, and then integrate x cos x dx = x sin x sin x dx = x sin x + cos x + C.

### Integration by trigonometric substitutions

This method involves replacing a variable or expression in the integrand with a trigonometric function or identity that simplifies the integration. For example, if we want to find (1 x^2) dx, we can substitute x = sin Î¸ and dx = cos Î¸ dÎ¸, and then integrate (1 sin^2 Î¸) cos Î¸ dÎ¸ = cos^2 Î¸ dÎ¸ = (Î¸/2) + (sin 2Î¸/4) + C.

### Integration by partial fractions

This method involves decomposing a rational function (a ratio of two polynomials) into simpler fractions that can be easily integrated. For example, if we want to find (x^2 + 1)/(x^3 + x) dx, we can write (x^2 + 1)/(x^3 + x) = A/x + B/(x + 1) + C/(x 1), where A, B and C are constants that can be found by equating coefficients. Then we can integrate (A/x + B/(x + 1) + C/(x 1)) dx = A ln x + B ln x + 1 + C ln x 1 + D.

Definite integrals and their properties: A definite integral of f(x) from a to b is the net area under the curve y = f(x) between x = a and x = b, denoted by af(x)dx. It can be calculated by finding an antiderivative F(x) of f(x) and then evaluating F(b) F(a). Definite integrals have some useful properties such as linearity, additivity, symmetry, comparison, etc.

Applications of integration: There are many practical problems that can be solved by using integration. Some of these problems involve finding:

### Area under a curve

The area under a curve y = f(x) between x = a and x = b is given by af(x)dx. For example, if we want to find the area under y = x^2 between x = 0 and x = 2, we can integrate 0x^2 dx = (x^3/3)0 = (8/3) (0/3) = (8/3).

### Length of a curve

The length of a curve y = f(x) between x = a and x = b is given by a(1 + (f'(x))^2) dx. For example, if we want to find the length of y = ln x between x = 1 and x = e, we can integrate 1(1 + (1/x)^2) dx 1.998.

### Volume and surface area of solids of revolution

A solid of revolution is formed by rotating a plane region around an axis. The volume and surface area of such solids can be found by using integration. For example, if we want to find the volume and surface area of a cone with base radius r and height h, we can consider rotating the region under y = (h/r)x between x = 0 and x = r around the x-axis. The volume is given by Ï€0(y^2) dx = Ï€0((h/r)^ Here is the continuation of the article. 2) dx = Ï€0(h/r)^2 x^2 dx = (Ï€h^2/3r^2)0x^2 dx = (Ï€h^2/3r^2)(x^3/3)0 = (Ï€h^2/9r)(r^3 0) = (Ï€h^2r/3). The surface area is given by Ï€0(y(1 + (y')^2)) dx = Ï€0((h/r)x(1 + (h/r)^2)) dx = (Ï€h/r)0x(1 + (h/r)^2) dx. To evaluate this integral, we can use the substitution u = x(1 + (h/r)^2) and du = (1 + (h/r)^2) dx. Then we get Ï€0x(1 + (h/r)^2) dx = Ï€0(u/(1 + (h/r)^2)) du = (Ï€/(1 + (h/r)^2))0u du = (Ï€/(1 + (h/r)^2))(u^2/2)0 = (Ï€/(1 + (h/r)^2))(u(r)^2/2 0) = (Ï€/(1 + (h/r)^2))(r^2(1 + (h/r)^2)/2) = (Ï€r/2)(r + (r^2 + h^2)). Adding the area of the base, we get the total surface area as Ï€r(r + (r^2 + h^2)) + Ï€r^2.

## Advanced Topics of Integral Calculus

The second part of the book covers some advanced topics of integral calculus that require more skills and knowledge. These topics include:

Integration of hyperbolic, rational, irrational and transcendental functions: These are some special types of functions that have their own rules and methods for integration. For example, to integrate hyperbolic functions, we can use their definitions in terms of exponential functions. To integrate rational functions, we can use partial fractions or long division. To integrate irrational functions, we can use trigonometric substitutions or rationalization. To integrate transcendental functions, we can use integration by parts or special formulas.

Reduction formulae: These are some general formulas that can help us find integrals of certain forms by reducing them to simpler integrals. For example, one reduction formula is sin^n x dx = (sin^(n1) x cos x)/n + ((n1)/n) sin^(n2) x dx, which can be used to integrate any power of sine function.

Beta and gamma functions: These are some special functions that are defined by integrals and have many applications in mathematics and science. The beta function is defined as B(x,y) = 0t^(x1)(1 t)^(y1) dt, and the gamma function is defined as Î“(x) = 0t^(x1)e^(t) dt. They are related by the formula B(x,y) = Î“(x)Î“(y)/Î“(x+y).

Integration of functions of two and three variables: These are integrals that involve more than one variable and can be used to find areas, volumes, masses, moments, centroids, etc. of plane and solid regions. For example, to find the area of a region R in the xy-plane, we can use a double integral Rf(x,y) dA, where f(x,y) is a function that gives the height of the region above the xy-plane. To find the volume of a solid S in space, we can use a triple integral Sf(x,y,z) dV, where f(x,y,z) is a function that gives the density of the solid.

### Conclusion

In this article, we have given you an overview of the book Integral Calculus by Hari Kishan pdf, which is a comprehensive guide for students who want to learn or improve their skills in integration. We have covered some of the basic and advanced concepts that are explained in the book, such as methods of integration, definite integrals, applications of integration, reduction formulae, beta and gamma functions, and integration of functions of two and three variables. We have also provided you with some examples and exercises to help you understand and practice these concepts.

If you are interested in reading this book, you can find the pdf version online or buy the print version from Amazon or other online stores. You can also check out some other books on integral calculus that we have listed below. We hope you have enjoyed this article and learned something new from it.

### Frequently Asked Questions

Where can I get the pdf version of the book?

You can download the pdf version of the book from this link: __https://www.ebookee.com/PDF-Integral-Calculus_Hari-Kishan_3527359.html__. You can also search for other sources online that offer free or paid downloads of the book.

How can I improve my skills in integral calculus?

The best way to improve your skills in integral calculus is to practice as much as possible. You can use the exercises and problems given in the book or other sources to test your understanding and application of the concepts. You can also watch online videos or lectures that explain the concepts in different ways. You can also join online forums or groups where you can ask questions or discuss with other students or experts.

What are some common mistakes to avoid in integration?

Some common mistakes to avoid in integration are:

Misapplying or forgetting the rules or formulas for integration.

Making algebraic or arithmetic errors while simplifying or evaluating expressions.

Mixing up indefinite and definite integrals.

Misusing or omitting constants of integration.

Misinterpreting or misapplying limits of integration.

Mischoosing or misusing methods or techniques of integration.

Miscalculating or mislabeling units or dimensions.

Misrepresenting or miscommunicating results or solutions.

To avoid these mistakes, you should always check your work carefully and review your steps logically. You should also use proper notation and terminology and show your work clearly and completely.

How can I check my answers for integration problems?

You can check your answers for integration problems by using different methods or techniques to solve them and comparing your results. You can also use online calculators or tools that can perform integration for you and verify your answers. You can also use differentiation to check your answers for indefinite integrals by applying the inverse relationship between differentiation and integration.

What are some other books on integral calculus that I can read?

Some other books on integral calculus that you can read are:

A Course in Calculus and Real Analysis, by Sudhir R. Ghorpade and Balmohan V. Limaye.

A First Course in Integral Equations, by Abdul-Majid Wazwaz.

An Introduction to Integral Calculus: A Textbook for Beginners, by Ravi P. Agarwal and Donal O'Regan.

Apostol's Calculus Volume II: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability , by Tom M. Apostol.

The Calculus Lifesaver: All the Tools You Need to Excel at Calculus , by Adrian Banner.

The Humongous Book of Calculus Problems , by W. Michael Kelley.

The Manga Guide to Calculus , by Hi Here is the continuation of the article. The Manga Guide to Calculus, by Hiroyuki Kojima and Shin Togami.