# Introduction to Antiderivative Its definition rules and calculations

What is antiderivative?, The general form of an Antiderivative, Rules of Antiderivative, Antiderivative of Trigonometric functions

## Introduction to Antiderivative Its definition, rules and calculations

The antiderivative is an easy concept but has importance in calculus easy concept of antiderivative is to check given function is the derivative of whose function. It can easily calculate by taking the integration of that function.

Anti-differentiation also plays a key role in physics position and is an antiderivative of velocity if you know the velocity throughout and also with the starting position you can easily calculate the position throughout the time.

Moreover, in this article basic definition of anti-derivative its formula, and methods to calculate it with the help of examples will be discussed.

### What is antiderivative?

“If G(x) is a function with G(x) = g(x), then here you can say that G(x) is an antiderivative of g(x) for all x in the domain of g”

There are two types of integration definite and indefinite. Antiderivatives are linked to integrals through the second fundamental theorem of calculus. it arises in the condition of rectilinear motion. Anti-difference is the discontinuous equivalent of the notion of the antiderivative.

Every continuous function has an antiderivative and infinitely many antiderivatives. If there are given two same functions to check their difference between antiderivatives see constants.

## Formula

∫ f(x) dx = F(x) + C

## The general form of an Antiderivative

Let G be an antiderivative of g over an interval I. Here,

• For each constant C, the function G(x) + C is also an antiderivative of g over I
• If F is an antiderivative of g over I, there is a constant C for which F (x) = G (x) + C over I.

## Introduction to Antiderivative Its definition calculations Rules of Antiderivative

There are a few main rules that are necessary to be followed while integrating a function to find its antiderivatives. These rules are discussed below:

### 1.    Difference Rule:

Antiderivative of a difference is equal to the difference of their antiderivatives. For better understanding, it can be expressed as

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

### 2.    Sum Rule

The Antiderivative of a sum is equal to the sum of their antiderivatives. For better understanding, it can be expressed as

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

### 3.    Constant function Rule

It is a common rule used in integration and also in derivation that a scalar can be taken out of an integral in this rule let’s suppose if p is a scalar then

∫ [P f(x)] dx = P ∫ [f(x)] dx

### 4.    Power Rule

It is also a common rule which is used in integration and it helps to solve integration expressions with radicals in them.

∫ [f(x)]n dx = [f(x)]n+1 / (n + 1) + C

## Antiderivative of Trigonometric functions

There are six basic trigonometric functions with which you are familiar like sine used as sin, cosine and used as cos, tangent used as tan, cosecant used as cosec, secant used as a sec, and cotangent used as cot where cosec, sec, and cot are reciprocal of sin cos and tan.

1. Antiderivative of cos(x) is equal to sin(x) + c where c is an arbitrary constant.

∫ [cos(x)] dx   = sin (x) + C

• Antiderivative of sin (x) is equal to – cos(x) + c where c is an arbitrary constant.

∫ [sin(x)] dx   = -sin (x) + C

• Antiderivative of tan (x) is equal to – ln|cos(x)| + C where c is an arbitrary constant.

∫ [tan(x)] dx   = -ln(cos(x)) + C

• Antiderivative of sec (x) is equal to ln |sec(x) + tan(x)|+ C where c is an arbitrary constant.

∫ [sec (x) dx   = ln |sec(x) + tan(x)|+ C

• Antiderivative of cosec (x) is equal to – ln |cosec(x) + cot(x)|+ C where c is an arbitrary constant.

∫ [cosec (x) dx   = – ln |cosec(x) + cot(x)|+ C

• Antiderivative of cot (x) is equal to ln|sin(x)| + C where c is an arbitrary constant.

∫ [cot (x) dx   = ln|sin(x)| + C

Similarly, for inverse trigonometric functions

1. ∫ [1/ (1 – x2) dx   = sin -1 (x) + C, where x ≠ ±1 and c is an arbitrary constant.
2. ∫ [-1/ (1 – x2) dx = cos -1 (x) + C, where x ≠ ±1 and c is an arbitrary constant.
3. ∫ [1/ (1 + x 2) dx   = tan -1 (x) + C, where c is an arbitrary constant.
4. ∫ [-1/ (1 + x2) dx = cot -1 (x) + C, where c is an arbitrary constant.
5. ∫ [-1/  (1 – x2) dx = cosec -1 (x) + C, where x ≠ ±1, 0 and c is an arbitrary constant.
6. ∫ [-1/ |x|√ (1 – x2) dx = sec -1 (x) + C, where x ≠ ±1, 0 and c is an arbitrary constant.

## Example Section

In this section with the help of examples antiderivative be evaluated:

Example 1:

Find the Anti-derivative of x3.

Solution:

f(x) = x3

With the help of a formula

∫ f(x) dx = F(x) + C, where c is an arbitrary constant.

f(x) = ∫ x3 dx

With the help of power rule

∫ [f(x)]n dx = [f(x)]n+1 / (n + 1) + C

F(x) = x3+1 / 3+1 + C

F(x)= x4/4 + C which is antiderivative of x3

To get rid of these calculations you can use online tools to get your results in a few seconds.

Example 2:

Find the derivative of sin(x)+cos(x).

Solution:

f (x) = sin(x) + cos(x)

f(x) = x3

With the help of a formula

∫ f(x) dx = F(x) + C, where c is an arbitrary constant.

f(x) = ∫ [sin(x) + cos(x)] dx

With the help of the sum rule

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

∫ [sin(x) + cos(x)] dx = ∫ [sin(x)] dx + ∫ [cos(x)] dx

∫ [sin(x) + cos(x)] dx = -cos(x) + sin(x) + C

F(x) = -cos(x) + sin(x) + C which is antiderivative of sin(x)+cos(x)

## Summary

In this article, basic definition with formula and rules for sum, difference, power, and constant are also discussed with the help of examples after reading and understanding this article one can easily defend this article. Moreover, its role in physics is working and with the help of this how can you calculate remaining things if few of them given also discussed.

Jessica Helen

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