For lecture.

1. Basic Calculus
Integration
Carl Phillip Jason C. Fulgencio

2. Integration
LEARNING OUTCOMES: At the end of the lesson, the
learner shall be able to:
1. Illustrate the antiderivative of a function;
2. Compute the general antiderivative of
polynomial functions;
3. Compute the general antiderivative of root
functions;
4. Compute the general antiderivative of
exponential functions; and,
5. Compute the general antiderivative of

3. Illustration of an
Antiderivative of a Function
Definition. A function F is an antiderivative of
the function f on an interval I if F’(x) = f(x)
for every value of x in I.
Theorem 1. If F is an antiderivative of f on an
interval I, then every antiderivative of f on I
is given by
F(x) + C
where C is an arbitrary constant.

4. Illustration of an
Antiderivative of a Function
Terminologies and Notations:
• Antidifferentation is the process of finding
the antiderivative.
• 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.

5. Illustration of an
Antiderivative of a Function
Terminologies and Notations:
• 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.

6. Illustration of an
Antiderivative of a Function
Example 1. Let f(x) = -8x3-10x+5 and F(x) = -2x4-
5x2+5x+3. Show that F(x) is an antiderivative of
f(x).
F’(x) = f(x)
Example 2. Determine if F(x) = x3 + x + 1, G(x)
= x3 + 2x + 1 or H(x) = x3 + x + 3 is/are
antiderivatives of f(x) = 3x2 + 1.
F’(x) = f(x) , G’(x) = f(x) and H’(x) = f(x)

7. Sample Problem
Determine the antiderivatives of the following
functions.
1. 𝑓 𝑥 = 8𝑥7 + 2𝑥3 − 1
2. 𝑓 𝑥 = −7
3. 𝑔 𝑥 = 2𝑥3
− 2𝑥 − 1
4. 𝑓 𝑥 = 9𝑥2 + 4𝑥

8. Antiderivative of Algebraic
Functions
Theorem 2. ∫𝒅𝒙 = 𝒙 + 𝑪
Theorem 3. ∫𝒂𝒇(𝒙)𝒅𝒙 = a∫𝒇 𝒙 𝒅𝒙 , where a is a
contant (any real number).
∫3𝑥2
𝑑𝑥 = 3∫𝑥2
𝑑𝑥
Theorem 4. ∫[𝒂𝒇 𝒙 ± 𝒂𝒈 𝒙 ]𝒅𝒙 = 𝒂∫𝒇(𝒙)𝒅𝒙 ± a∫g(x)dx
∫(3𝑥2
+7)𝑑𝑥 = 3∫𝑥2
𝑑𝑥 + 7∫𝑑𝑥

9. Antiderivative of Algebraic
Functions
Theorem 5. The Power Rule for
Antidifferentiation
if n is any real number, then
∫𝒙𝒏
𝒅𝒙 =
𝒙𝒏+𝟏
𝒏+𝟏
+ 𝒄 ; 𝒏 ≠ −𝟏
*** ang ay okay lang maging negative (ex. -2,-3,-3,
-2/3 etc…), pero hindi pweding -1 ang value ng n,
dahil magiging 0 ang denominator,

10. Theorem 2 (∫𝑑𝑥 = 𝑥 + 𝐶 ). Example
1. ∫4𝑑𝑥
2. ∫ − 15𝑑𝑥
3. ∫ − 2𝑑𝑥
4. ∫1,000,000.00𝑑𝑥

11. Theorem 3 (∫𝑎𝑓(𝑥)𝑑𝑥 = a∫𝑓 𝑥 𝑑𝑥 ).
Example
1. ∫9𝑥𝑑𝑥
2. ∫ − 15𝑥𝑑𝑥
3. ∫5 𝑥𝑑𝑥
4. ∫103
𝑥𝑑𝑥

12. Theorem 4. Example
∫[𝒂𝒇 𝒙 ± 𝒂𝒈 𝒙 ]𝒅𝒙 = 𝒂∫𝒇(𝒙)𝒅𝒙 ± a∫g(x)dx
1. ∫(10𝑥4 + 2𝑥2)𝑑𝑥
2. ∫ 16𝑥3 − 3𝑥2 − 1 𝑑𝑥
3. ∫(𝑥2 − 4)(𝑥 + 1)𝑑𝑥
4. ∫(𝑥2 − 4)( 𝑥 + 𝑥)𝑑𝑥

13. Theorem 5. Example
∫𝒙𝒏
𝒅𝒙 =
𝒙𝒏+𝟏
𝒏 + 𝟏
+ 𝒄
1. ∫𝑥4𝑑𝑥
2. ∫5𝑥2
𝑑𝑥
3. ∫6 𝑥𝑑𝑥
4. ∫
3
𝑥2𝑑𝑥

14. Assignment
(Due Date April 29,2024)
1. 𝑥3 𝑑𝑥
2. 4
7
𝑥4𝑑𝑥
3. (4𝑥 − 5)𝑑𝑥
4. (3𝑥 − 9)𝑑𝑥
5. (5𝑥4
−8𝑥3
+ 9𝑥2
− 2𝑥) 𝑑𝑥
6.
1
𝑥5 𝑑𝑥
7.
2𝑥4−𝑥
𝑥3 𝑑𝑥
8.
5𝑡2+7
𝑡
𝑑𝑡
9. 𝑥(𝑥 +
1
𝑥
)𝑑𝑥
10. 𝑥(x2 − 3𝑥 + 1)𝑑𝑥