Integral as an antiderivative

What is an antiderivative?

As the name suggests, an antiderivative is the opposite of differentiation. If we want to know the derivative of [x2] we simply use the formula d/dx [x2] and get our required answer as 2x.

But, the derivative of [x2 + 1] is also x2, the derivative of [x2 + ∏] is also x2.

So what to do when we want to know whose derivative 2x is? This is when antiderivative comes into play.

Antiderivative is written by using the elongated ‘s’ symbol that we commonly used for derivatives and writing dx around the function we want the antiderivative of. 

For Example-

So, Antiderivative of 2x is f 2x dx= x²+c. 

f 2x dx

This part of the equation is called the ‘Indefinite Integral’ of 2x which is just another way of saying ‘The antiderivative of 2x’

In simple words, an antiderivative is the opposite of a derivative. It’s also known as the integral, and often refers to the area under a graph (graphically).

Basic Antiderivatives

The symbol used to represent an antiderivative is the elongated ‘S’ symbol- ∫.

f(x) dx

∫ (x)dx implies the antiderivative of ‘x’ from ‘f’. If F is an antiderivative of f, then we can simply write that f (x)dx = F + c. 

Here, c is called the ‘Constant of Integration’.

A single continuous function has an infinite number of antiderivatives, we can instead of saying “too many antiderivatives”, a ‘family of antiderivatives’, each of which differs from one another by a constant. 

So, if the function F is an antiderivative of ‘f’, then G = F + c can also be considered as an antiderivative of ‘f’, and F and G are to be considered in the same family of antiderivatives.

Antiderivatives are informally connected with areas. 

NOTE: If F is an antiderivative of ‘f’, then the function f has a whole family of antiderivatives. Each antiderivative of ‘f’ is the sum of F and some constant C.

To calculate integrals, we must know that it is very important to find antiderivatives of functions. 

Antiderivative Formula

It is very important to know that antiderivatives are not distinctive. A given function can have countless antiderivatives.

One must remember that two antiderivatives of a given function are different from each other due to a constant. This provides aid to us for writing general formulae for any antiderivative.

An indefinite integral is an integral which has no or zero terminals; it simply asks us to find a general antiderivative of the function. It is not a single function but a group of functions, which are different from each other because of the existence of constants.

A definite integral is one which is written with the help of terminals. It also helps us to measure the area under a graph from one terminal to another.

Common formulae used for performing antiderivative solutions are-

f(x^n) dx= x^n+1/n+1 +C, n=-1

f(1/x) dx = ln(|x|) + C

 f(e ^ x) dx = e ^ x + C

 f( sin x) dx = – cos x + C

 f( cos x) dx = sin x + C

 f(sec² x) dx = tan x + C

 f( sec(x) * tan x) dx = sec(x) + C

 f( cosec² * x) dx = – cot x + C

 f( cot x * csc(x)) dx = – csc(x) + C

 f( 1/(x ^ 2 + 1)) dx = arctan(x) + C

f( 1/(√(1 – x²))) dx = arcsin(x) + C

 f( 1/(√( x²-1)) dx = arcsec(x) + C

Rules for antiderivative 

In the field of Mathematics ‘rules of antiderivatives’ are a set of basic important rules which help the student to find antiderivative values of different problems.

As we all know now, antiderivatives is the opposite of differentiation. Antiderivative rules help us to find antiderivatives of subtraction (difference) or addition(sum) of functions, multiplication and division of functions, their multiples in scalar form and the constants of a function.

antiderivative rules enable us to make the process of solving the antiderivatives easy and also make sure that the students follow a form of discipline.

Some of the commonly used rules of antiderivative for multiplication, division, addition, subtraction,  are:

1. Antiderivative Rule of Product
2. Antiderivative Quotient Rule of Quotient
3. Antiderivative Chain Rule
4. Antiderivative Rule of Power
5. Antiderivative Rule of Function for Sum and Difference
6. Antiderivative Rule of Functions for Scalar Multiple

Antiderivative Power Rule

To find antiderivatives of basic functions, the following rules can be used:

xndx = xn+1 + c

 as long as n does not equal -1.

Antiderivative Chain Rule

The antiderivative chain rule is used if the integral is of the form ∫u'(x) f(u(x)) dx. It is commonly known as the u-substitution method of anti-differentiation. We generally substitute the function u(x) by assuming it to be another variable.

Antiderivative Product Rule

The formula for the antiderivative product rule is

 ∫f(x).g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) [ ∫g(x) dx)]dx + C.

Antiderivative Division Rule

A different way to determine the antiderivative of the division of functions is, by considering a function of the form f(x)/g(x). Now, differentiating this we get our desired rule. We have the antiderivative rule for quotient as:

∫d(u/v) = u/v + ∫[u/v2] dv

Antiderivative Rule of Function for Scalar Multiple

In order to know antiderivatives of a given scalar multiple of a given function f(x), it can be found with the help of the known formula, 

∫ kf (x) dx = k ∫f(x) dx. 

This formula tells us that, the anti-differentiation of kf(x) is = to k times the anti-differentiation of ‘f’(x), where k is considered a scalar.

Addition and Subtraction Antiderivative Rule

This rule is one of the most common antiderivative rules. When the anti-differentiation of the addition and subtraction of functions is to be known, the following formulae are used-

∫ [ f (x) – g (x) ] dx = ∫ f (x) dx – ∫ g (x) dx

∫ [ f (x) + g (x) ] dx = ∫ f (x) dx + ∫ g (x) dx