Antiderivatives: The Antiderivative of a function is the inverse of the derivative of the function. Antiderivative is also called the Integral of a function. Suppose the derivative of a function d/dx[f(x)] is F(x) + C then the antiderivative of [F(x) + C] dx of the F(x) + C is f(x). An example explains this if d/dx(sin x) is cos x then, the antiderivative of cos x, given as ∫(cos x) dx is sin x.
Antiderivative of any function is used for various purposes, to give the area of the curve, to find the volume of any 3-D curve, and others. In this article, we will learn about, Antiderivatives, Antiderivatives Formulas, Antiderivatives rules, and others in detail.
What are Antiderivatives?
Antiderivative of a function is the reverse operation of the function’s derivative. Mathematically, we also call Antiderivative the Integral of a function.
Suppose, the derivative of the function F(x) is,
F'(x) = f(x)
For all x in the domain of f. If f(x), is the derivative of F(x) then the antiderivative or integral of the function f(x) is,
This can be explain by the example lets take a function f(x) = x2, on differentiating this function, the output is another function g(x) = 2x.
For, g(x) = 2x the anti-derivative will be,
f(x) = x2
d/dx[f(x)] = f'(x) = g(x)
g(x) = 2x
Now the antiderivative of 2x is,
= ∫g(x).dx
= ∫(2x).dx
= 2(x2)/2 + C
= x2 + C
Here the symbol ∫ denotes the anti-derivative operator, it is called indefinite integrals. Also, C is integration constant, or Antiderivative constant. Antiderivatives are classified into two types,
- Indefinite Antiderivatives
- Definite Antiderivatives
Indefinite Antiderivative
Indefinite antiderative also called the indefinite integral is anti derivative of that function in which the limit of the antiderivative (integration) is not given and the result is accompanied with a constant value (generally C) called the constant of integration. Suppose we have a function F(x) whose derivative is f(x) then,
∫ f(x) dx = F(x) + C
where C is Integration Constant
Definite Antiderivative
Definite Antiderivative or definite integral is the antiderivative of any function inside a closed interval. In this integration the constant of integration is not present and the answer to the integration is some contact value. Suppose we have a function F(x) that is defined on closed interval [a, b] then if its derivative is f(x), its definite antiderative is written as,
∫ab f(x) = [F(x)]ab = F(b) – F(a)
Rules of Antiderivative
Vrious rules that are used to easily solve problems of Antiderivaties are,
Constant Rule
∫kf(x)dx = k ∫ f(x)dx, here “k” is any constant.
Sum Rule
This rule states that the integral of sum of two functions is equal to sum of integrals of those two functions.
∫(f(x) + g(x))dx = ∫ f(x)dx + ∫g(x)dx
Difference Rule
This rule states that the integral of difference of two functions is equal to difference of integrals of those two functions.
∫(f(x) – g(x))dx = ∫ f(x)dx – ∫g(x)dx
Properties of Antiderivatives
Antiderivative of a function has various properties and the important properties of Antiderivative are,
- ∫-f(x)dx = -∫f(x)dx
- ∫ f(x) dx = ∫g(x) dx if ∫[f(x) – g(x)]dx = 0
- ∫ [k1f1(x) + k2f2(x) + …+knfn(x)]dx = k1∫ f1(x)dx + k2∫ f2(x)dx + … + kn∫ fn(x)dx
Vrious formula used for finding the antiderivative of the functions are,
- ∫ xn dx = x(n + 1)/(n + 1) + C
- ∫ ex dx = ex + C
- ∫ 1/x dx = log |x| + C
Learn More, Integration Formulas
Calculation of Antiderivative of a Function
It is not always possible to just guess the integral of any function by thinking of the reverse differentiation process. A formal approach or a formula is necessary for calculating the Antiderivatives.
To calculate the antiderivative of any function follow the steps added below,
Check the given integral and try to guess the derivative of the function whose antiderivative is to be calculated.
Example: Find the antiderivative of xn.
Solution:
Antiderivative of xn = ∫ xn dx
Using Integration Formulas
= x(n+1)/(n+1) {except when n = -1}
The table below represents some standard functions and their integrals.
| Function | Integral |
|---|
| sin(x) | -cos(x) + C |
| cos(x) | sin(x) + C |
| sec2(x) | tan(x) + C |
| ex | ex + C |
| 1/x | ln(x) + C |
Antiderivative of Trigonometric Functions
Antiderivative of the trigonometric fuctions is easily found that helps us to solve various problems of integration. Antiderivative of the Trigonometric Functions are,
- ∫ sin x dx = -cos x + C
- ∫ cos x dx = sin x + C
- ∫ tan x dx = -ln |cos x| + C = ln |sec x| + C
- ∫ cot x dx = ln |sin x| + C = -ln |cosec x| + C
- ∫ sec x dx = ln |sec x + tan x| + C
- ∫ cosec x dx = – ln |cosec x + cot x| + C
- ∫ cos (ax + b)x dx = (1/a) sin (ax + b) + C
- ∫ sin (ax + b)x dx = -(1/a) cos (ax + b) + C
Antiderivative of Inverse Trig Functions
There are some functions whose antiderivative gives Inverse Trigonometric Functions that are,
- ∫ 1/√(1 – x2).dx = sin-1x + C
- ∫ 1/(1 – x2).dx = -cos-1x + C
- ∫ 1/(1 + x2).dx = tan-1x + C
- ∫ 1/(1 + x2).dx = -cot-1x + C
- ∫ 1/x√(x2 – 1).dx = sec-1x + C
- ∫ 1/x√(x2 – 1).dx = -cosec-1x + C
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Examples on Antiderivatives
Example 1: Find the integral for the given function,
f(x) = 2x + 3
Solution:
Using Integral Formula,
[Tex]\int x^ndx = \frac{x^{n+1}}{n+1} + c
[/Tex]
Given,
f(x) = 2x + 3
[Tex]\int(2x + 3)dx
[/Tex]
Splitting the function
[Tex]\int(2x + 3)dx
[/Tex]
⇒[Tex]\int 2xdx + \int3dx
[/Tex]
⇒[Tex]2\int xdx + 3\int 1dx
[/Tex]
⇒[Tex]2\frac{x^2}{2} + 3x + C
[/Tex]
⇒ x2 + 3x + C
Example 2: Find the integral for the given function,
f(x) = x2 – 3x
Solution:
Using Integral Formula,
[Tex]\int x^ndx = \frac{x^{n+1}}{n+1} + c
[/Tex]
Given,
f(x) = x2 – 3x
[Tex]\int(x^2 – 3x)dx
[/Tex]
Splitting the function
[Tex]\int(x^2 – 3x)dx
[/Tex]
⇒[Tex]\int x^2dx – 3\int xdx
[/Tex]
⇒[Tex]\frac{x^3}{3} – \frac{3x^2}{2} + C
[/Tex]
Example 3: Find the integral for the given function,
f(x) = x3 + 5x2 + 6x + 1
Solution:
Using Integral Formula,
[Tex]\int x^ndx = \frac{x^{n+1}}{n+1} + c
[/Tex]
Given,
f(x) = x3 + 5x2 + 6x + 1
[Tex]\int(x^3 + 5x^2 + 6x + 1)dx
[/Tex]
Splitting the function
[Tex]\int(x^3 + 5x^2 + 6x + 1)dx
[/Tex]
⇒[Tex]\int x^3dx + \int 5x^2dx + \int 6xdx + \int 1dx
[/Tex]
⇒[Tex]\frac{x^4}{4} + \frac{5x^3}{3}+ 3x^2 + x
[/Tex]
Example 4: Find the integral for the given function,
f(x) = sin(x) – cos(x)
Solution:
Using Integral Formula,
Given,
f(x) = sin(x) – cos(x)
[Tex]\int(sin(x) – cos(x))dx
[/Tex]
Splitting the function
[Tex]\int(sin(x) – cos(x))dx
[/Tex]
⇒[Tex]\int sin(x)dx – \int cos(x)dx
[/Tex]
⇒[Tex]-cos(x) – sin(x) + C
[/Tex]
Example 5: Find the integral for the given function,
f(x) = 2sin(x) + sec2(x) + 7ex
Solution:
Given,
f(x) = 2sin(x) + sec2(x) + 7ex
[Tex]\int(2sin(x) + sec^2(x) + 7e^x)dx
[/Tex]
Splitting the function
[Tex]\int(2sin(x) + sec^2(x) + 7e^x)dx
[/Tex]
⇒[Tex]\int 2sin(x)dx + \int sec^2(x)dx + \int 7e^xdx
[/Tex]
⇒[Tex]2\int sin(x)dx + \int sec^2(x)dx + 7\int e^xdx
[/Tex]
⇒[Tex]-2cos(x) + tan(x)dx + 7e^x + C
[/Tex]
Example 6: Find the integral for the given function,
f(x) = [Tex]\frac{x – 3}{x}
[/Tex]
Solution:
Using Integral Formula,
[Tex]\int x^ndx = \frac{x^{n+1}}{n+1} + c
[/Tex]
Given,
f(x) = [Tex]\frac{x – 3}{x}
[/Tex]
[Tex]\int(\frac{x – 3}{x})dx
[/Tex]
Splitting the function
[Tex]\int(1 – \frac{3}{x})dx
[/Tex]
⇒[Tex]\int1dx – \int \frac{3}{x}dx
[/Tex]
⇒ x – 3ln(x) + C
Example 7: Find the integral for the given function,
f(x) = x2 – 4x + 4
Solution:
Using Integral Formula,
[Tex]\int x^ndx = \frac{x^{n+1}}{n+1} + c
[/Tex]
Given,
f(x) = x2 – 4x + 4
[Tex]\int(x^2 – 4x + 4)dx
[/Tex]
Splitting the function
[Tex]\int x^2dx – \int 4xdx + \int 4dx
[/Tex]
⇒[Tex]\int x^2dx – 4\int xdx + 4\int dx
[/Tex]
Antiderivatives Worksheet
Q1: ∫1/√x dx
Q2: ∫a2logax dx
Q3: ∫2/(1 + cos 2x)dx
Q4: ∫3x+3dx
Q5: ∫1/2tan x dx
Summary
Antiderivatives, also known as indefinite integrals, are the reverse process of differentiation. Given a function f(x), an antiderivative F(x) is a function whose derivative is f(x), i.e., F′(x)=f(x). Antiderivatives are crucial in solving problems involving integration, as they provide a means to determine the original function from its rate of change. The process of finding an antiderivative involves determining the function F(x) plus a constant of integration, often denoted as C, because differentiation of a constant is zero and thus does not affect the derivative.
FAQs on Antiderivatives
What is Antiderivative in simple terms?
Andtiderivative as the name suggest is the inverse operation of derivative of any function. Antiderivative is also called Integration. Suppose the derivative of any function f(x) is F(x) then the anti derivative of F(x) is f(x) + C where, C is integration constant.
Are Antiderivatives Same as Integrals?
Yes, antiderivative are similar to integral. They are equivalent in mathematics and can be used alternatively.
What is the Power Rule for Antiderivatives?
Power rule in antiderivative states that antiderivative of a function xn (where the value of one is never equal to -1) is given using the formula,
∫ xn dx = x(n + 1)/(n + 1) + C
What is the Antiderivative of 1 / x?
Antiderivative of 1 / x is given by the formula,
∫1/x.dx = ln|x| + C
What is Antiderivative of Sin x?
Antiderivative of Sin x is,
∫sin x.dx = -cos x + C
What is Antiderivative of Cos x?
Antiderivative of Cos x is,
∫cos x.dx = sin x + C
What is Antiderivative of Tan x?
Antiderivative of Tan x is,
∫tan x.dx = ln |sec x| + C