This presentation is for the Grade 11 STEM Basic Calculus

1. ANTIDERIVATIVE
OF A FUNCTION

2. OBJECTIVES
 Illustrate an antiderivative of a function
 Compute the general antiderivative of polynomial, radical,
exponential, and trigonometric function

3. GENERALIZATION
In the previous discussion, we learned how to find the
derivatives of different functions. Now, we will introduce
the inverse of differentiation. We shall call this process
as antidifferentiation.
A natural question then arises:
a. Given a function f, can we find a function F whose
derivative is f?
b. Is it possible to find a function 𝒚 = 𝑭(𝒙) for which
f(x) is the derivative?

4. GENERALIZATION
This is the “anti” or inverse problem of finding the
derivative. Thus the function F(x) is called an
antiderivative of f(x) if and only if F’(x) = f(x).

5. GENERALIZATION
F(x) is an integral of f(x) with respect to x if
and only if F(x) is an antiderivative of f(x). That is,
F(x) = ∫ 𝑓(𝑥)𝑑𝑥 if and only if F’(x) = f(x)
Definition of Integral

6. GENERALIZATION
 Antidifferentiation is the process of finding the
antiderivative.
 The symbol ∫ , also called the integral sign,
denotes the operation of antidifferentiation.
 The function f is called integrand. • If F is an
antiderivative of f, we write ∫ 𝒇(𝒙)𝒅𝒙 = 𝑭(𝒙) + 𝑪.
TERMINOLOGIES AND NOTATIONS:

7. GENERALIZATION
 The symbols ∫ and dx go hand-in-hand and dx
helps to 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.
TERMINOLOGIES AND NOTATIONS:

8. GENERALIZATION
(a) If k is a constant, ∫ 𝑘𝑑𝑥 = 𝑘𝑥 +
BASIC THEOREMS ON ANTIDIFFERENTIATION

9. GENERALIZATION
(b) If n is any real number and n ≠ −1. 𝑡ℎ𝑒𝑛 ∫ 𝑥𝑛𝑑𝑥
=
𝑥𝑛+1
𝑛+1
+ C where x is a differentiable
function.
BASIC THEOREMS ON ANTIDIFFERENTIATION

10. GENERALIZATION
(c) The integral of a sum of functions is the sum of
the integrals of the functions. ∫(𝑓(𝑥) + 𝑔(𝑥))𝑑𝑥
= ∫ 𝑓(𝑥)𝑑𝑥 + ∫ 𝑔(𝑥)𝑑𝑥
BASIC THEOREMS ON ANTIDIFFERENTIATION

11. GENERALIZATION
(d) The integral of the product of a constant, a,
and a function, f(x), is the product of the
constant and the integral of the function.
∫ 𝑎𝑓(𝑥)𝑑𝑥 = 𝑎 ∫ 𝑓(𝑥)𝑑𝑥.
BASIC THEOREMS ON ANTIDIFFERENTIATION

12. EXAMPLES
1. Find the integral of ∫ 1𝑑𝑥 = ∫ 𝑑𝑥
SOLUTION:
∫ 𝑘𝑑𝑥 = 𝑘𝑥 + C
∫ 1𝑑𝑥 = 𝑥 + C

13. EXAMPLES
2. Find the integral of ∫ 5 𝑑𝑥.
SOLUTION:
∫ 𝑘𝑑𝑥 = 𝑘𝑥 + C
3. Find the integral of ∫ 3 𝑑𝑥.

14. EXAMPLES
4. Find the integral of ∫ 𝑥2 𝑑𝑥.
SOLUTION:
∫ 𝑥𝑛𝑑𝑥 =
𝑥𝑛+1
𝑛+1
+ C
5. Find the integral of ∫ 𝑥5 𝑑𝑥.

15. EXAMPLES
6. Find the antiderivative of ∫(4𝑥+7)dx
SOLUTION:
∫(𝑓(𝑥) + 𝑔(𝑥))𝑑𝑥 = ∫ 𝑓(𝑥)𝑑𝑥 + ∫ 𝑔(𝑥)𝑑𝑥

16. EXERCISE
Determine the antiderivative of the following functions.
1. Evaluate ∫(3𝑥 + 2)𝑑x
2. Evaluate ∫(12𝑥2 + 2𝑥)𝑑x