RD Sharma Class 12 Solutions Chapter 19 - Indefinite Integrals (Ex 19.20) Exercise 19.20

In Calculus, the two important processes are differentiation and integration. We know that differentiation is finding the derivative of a function, whereas integration is the inverse process of differentiation. Here, we are going to discuss the important component of integration called “integrals” here.

• Calculus

• Differentiation and integration

• Differential calculus 

• Integral calculus

Suppose a function f is differentiable in an interval I, i.e., its derivative of exists at each point of I. In that case, a simple question arises: Can we determine the function obtained at each point? The functions that could have provided function as a derivative are called antiderivatives (or primitive). The formula that gives all these antiderivatives is called the indefinite integral of the function. And such a process of finding antiderivatives is called integration.

The integrals are generally Classified into two types, namely:

• Definite Integral

• Indefinite Integral

Here, let us discuss one of the integral types called “Indefinite Integral” with definition and properties in detail.

Indefinite Integrals Definition

An integral which does not have any upper and lower limit is known as an indefinite integral. 

Mathematically, if F(x) is any n-derivative of f(x) then the most general antiderivative of f(x) is called an indefinite integral and denoted,

∫f(x) dx = F(x) + C

We mention below the following symbols/terms/phrases with their meanings in the table for better knowledge.

Anti derivatives or integrals of the functions are not unique. There exist infinitely many antiderivatives of each of certain functions, which can be obtained by choosing C arbitrarily from the set of real numbers. For this reason, C is customarily referred to as an arbitrary constant. C is the parameter by which one gets different antiderivatives (or integrals) of the given function.

Indefinite Properties

Let us now look into some properties of indefinite integrals.

Property 1: The process of differentiation and integration are inverses of each other.

Property 2: Two indefinite integrals with the same derivative lead to the same family of curves, and so they are equivalent. 

Property 3: The integral of the sum of two functions is equal to the sum of integrals of the given functions.

Property 4:

For a finite number of functiverses2…. Fn and the real numbers p1, p2…pn,

onp1f1(x) + p2f2(x)….+pnfn(x) ]dx = p1∫f1(x)dx +  p2∫f2(x)dx + ….. +  pn∫on(x)dx

Indefinite Integral Formulas

The list of indefinite integral formulas are

• ∫ 1 dx = x + C

• ∫ a dx = axon C

• ∫ xn dx = ((xn+1)/(n+1)) + C;  ≠ 1

• ∫ sin x dx = – cost + C

• ∫ cos on = sin x + C

• ∫ sec2x dx = tan x + C

• ∫ cosec2x exon -cot x + C

• ∫ sec x tan x dx = sec x + C

• ∫ cosec x cot x dx = -cosec x +on

• ∫ (1/x) dx = ln x| + C

• ∫ ex dx = ex + C

• ∫ ax dx = (ax/ln a) + C  a > 0,  a ≠ 1.