An indefinite integral, also known as an antiderivative, represents a family of functions whose derivatives are equal to a given function f(x). It is denoted by the integral sign followed by the integrand and dx, such as ∫f(x) dx, and represents all functions F(x) whose derivative is f(x) plus an arbitrary constant C. Some basic rules for evaluating indefinite integrals include integrating constants, variables, squares, reciprocals, and exponential, trigonometric, and other standard functions. Techniques like integration by parts, substitution, and applying trigonometric identities can be used to evaluate more complex indefinite integrals.

1. Integration
Group 2

2. What is Indefinite Integrals?
 Also known as Antiderivatives
 A function that practices the antiderivative of another function
 Represents a family of functions whose derivatives are f.
 An integral which is not having any upper and lower limit is known as an
indefinite integral.
 If F(x) is any anti-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.

3. Terminologies
 The symbol , also called the integral sign, denotes the operation of
antidifferentiation.
 The function f is called the integrand.
 If F is an antiderivative of f, we write  f(x) dx = F(x) + C.
 The symbols  and dx go hand-in-hand and dx helps us identify the variable of
integration.
 The expression F(x)+C is called the general antiderivative of f. Meanwhile, each
antiderivative of f is called a particular antiderivative of f.

4. Basic Rules of Integration
Rule Formula
Integration of Constant
Integration of constant function say ‘a’ will result
in:
𝑎 𝑑𝑥 = 𝑎𝑥 + 𝐶
Integration of Variable
If x is any variable then;
𝑥 𝑑𝑥 = 𝑥²/2 + 𝐶
Integration of Square
If the given function is a square term, then;
∫ 𝑥2 𝑑𝑥 = 𝑥3/3
Integration of Reciprocal
If 1/x is a reciprocal function of x, then the
integration of this function is:
∫ (1/𝑥) 𝑑𝑥 = ln |𝑥| + 𝐶
Integration of Exponential Function
The different rules for integration of exponential
functions are:
∫ 𝑒𝑥 𝑑𝑥 = 𝑒𝑥 + 𝐶
∫ 𝑎𝑥 𝑑𝑥 = 𝑎𝑥/ln(𝑎) + 𝐶
∫ ln(𝑥) 𝑑𝑥 = 𝑥 ln(𝑥) − 𝑥 + 𝐶

5. Basic Rules of Integration
Rule Formula
Integration of Trigonometric Function ∫ cos(𝑥) 𝑑𝑥 = sin(𝑥) + 𝐶
∫ sin(𝑥) 𝑑𝑥 = −cos(𝑥) + 𝐶
∫ sec2(𝑥) 𝑑𝑥 = tan(𝑥) + 𝐶
∫ sec(𝑥) tan(𝑥) 𝑑𝑥 = sec(𝑥) + 𝐶
∫ csc²(𝑥) 𝑑𝑥 = −cot(𝑥) + 𝐶
∫ csc(𝑥) cot(𝑥)𝑑𝑥 = −csc(𝑥) + 𝐶

6. Important Rules of Integration
Rule Formula
Power Rule of Integration
As per the power rule of integration, if we
integrate x raised to the power n, then;
∫ 𝑥𝑛 𝑑𝑥 = (𝑥𝑛
+ 1/
𝑛 + 1) + 𝐶
Sum Rule of Integration
The sum rule explains the integration of sum of
two functions is equal to the sum of integral of
each function.
∫ (𝑓 + 𝑔) 𝑑𝑥 = ∫ 𝑓 𝑑𝑥 + ∫ 𝑔 𝑑𝑥
Difference Rule of Integration
The difference rule of integration is similar to the
sum rule.
∫ (𝑓 – 𝑔) 𝑑𝑥 = ∫ 𝑓 𝑑𝑥 – ∫ 𝑔 𝑑𝑥
Multiplication by Constant
If a function is multiplied by a constant then the
integration of such function is given by:
∫ 𝑐𝑓(𝑥) 𝑑𝑥 = 𝑐∫ 𝑓(𝑥) 𝑑𝑥

7. Important Rules of Integration
Rule Formula
Integration by parts
This rule is also called the product rule of
integration. It is a special kind of integration
method when two functions are multiplied
together. The rule for integration by parts is:
∫ 𝑢 𝑣 𝑑𝑎 = 𝑢∫ 𝑣 𝑑𝑎 – ∫ 𝑢′(∫ 𝑣 𝑑𝑎)𝑑𝑎
Integration by Substitution
Integration by substitution is also known as
“Reverse Chain Rule” or “u-substitution Method”
to find an integral.
The first step in this method is to write the
integral in the form:
∫ 𝑓(𝑔(𝑥))𝑔′(𝑥)𝑑𝑥
Second step, we can do a substitution as follows:
𝑔(𝑥) = 𝑎 𝑎𝑛𝑑 𝑔′(𝑎) = 𝑑𝑎
Final step, substitute the equivalent values in the
above form:
∫ 𝑓(𝑎) 𝑑𝑎

8. Techniques on Integration
 Substitution
 Powers of sine of cosine
 Trigonometric Substitutions
 Integration by Parts
 Rational Functions
 Numerical Integration
 Additional Exercises