Integral Calculus Study Notes for GATE CSE
By BYJU'S Exam Prep
Updated on: September 25th, 2023
Integral Calculus Study Notes- INTRODUCTION
Integration is the reverse process of differentiation. It is sometimes called anti-differentiation. The topic of integration is often approached in several alternative ways. Perhaps the only way of introducing it’s to consider it as differentiation in reverse.
Table of content
Differentiation in Reverse (Anti-Derivative)
Suppose we differentiate the function F(x) = 3x2 +7x-2. We obtain its derivative as.
This process is illustrated in Figure 1.
Figure 1
In this case, we can say that the derivative of F(x) = 3x2 +7x-2. is equal to 6x+7 . However, there are many other functions that also have a derivative 6x+7. Some of these are 3x2 +7x+3, 3x2 +7x,3x2 +7x-11 and so on. The reason why all of those functions have an equivalent derivative is that the constant term disappears during differentiation. So, all of these are anti-derivatives. Given any anti-derivative of f(x), all others are often obtained by simply adding a special constant. In other words.
if F() is an anti-derivative of, then so too is F(x)+C for any constant and this actually describes the definition of Indefinite Integration.
INDEFINITE INTEGRATION
We call the set of all anti-derivatives of a function because the integral of the function. . The indefinite integral of the function f(x) is written as
and read as he indefinite integral of f(x) with respect to x. The function f(x) that is being integrated is called the integrand, and the variable x is called the variable of integration and the C is called the constant of integration.
Properties of the Indefinite Integral
Basically, there are three properties of anti-derivatives which been applied so as to unravel the mixing for any quiet functions.
Integral of Polynomial Functions
Properties of the Integral of Polynomial Functions
Example:
Integral of Exponential Functions
The formula of the Integral of Exponential Functions
Example:
Integral of Logarithmic Functions
The formula of Integral of Logarithmic Functions
DEFINITE INTEGRATION
In this section, the concept of “definite integrals” is introduced which can link the concept of area to other important concepts like length, volume, density, probability, and work.
Based on the Figure, the curve f(x) s is nonnegative and continuous on an interval [a,b]. The area of which is under A the graph of f(x) over the interval [a,b] can be represented by the definite integral.
Note that there’s no constant in integral, therefore integral is usually in number.
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