CALCULUS

CHAPTER 4
INTEGRAL CALCULUS

A
n

a
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tiderivative of a function or integration is the opposite of derivative or
differentiation which undo each other as like multiplication is opposite of
division. Bothe integral and derivative are opposites of each other which are
known as the Fundamental Theorem of Calculus.
An integral calculus deals with the theory and applications of integrals, the
total size or value, such as lengths, areas and volumes. It is a function whose
rate of change, or derivative, equals the function being integrated.
An example of integration is a velocity function yields a distance traveled by
an object over an interval of time to be calculated. As a result, much of
integral calculus is the derivation of formulas for finding antiderivatives. The
great utility of the subject emanates from its use in solving differential
equations.

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Topic: ANTIDERIVATIVE OF FUNCTION

Time Frame: 6 days

OBJECTIVE

Define the antiderivative of a function.

Evaluate the indefinite integral of a given function.

LOOKING BACK

WHERE IT CAME FROM

Description: This activity will serve as a spring board to understand
the process of anti-differentiation.
Directions: From the pool of expressions below, choose the
appropriate functions that will corresponds to the derivatives
describe in each item.

𝑥 2 +5𝑥 +6
𝑦 = 𝑥4 𝑦 = 𝑥2 − 8 𝑓 (𝑥) = 𝑥 2 −2

𝑦 = (2𝑥 + 1)3 (4𝑥 − 1)2 𝑓 (𝑥) = 4𝑥 3 − 7𝑥 2 + 7𝑥 + 3

𝑓 (𝑥 ) = √𝑥 𝑓 (𝑥 ) = 1/2 𝑓 (𝑥 ) = √3𝑥 + 2
1
𝑦 = (5𝑥 2 −11)6 𝑦 = (3 + 5𝑥)2

1.

3. 𝑓′(𝑥) = 0 8. 𝑦′ = 30 + 50𝑥

10. 𝑓′(𝑥) = 12𝑥2 − 14𝑥 + 7

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LET’S GET STARTED

THE IF – THEN GAME

Description: The goal of this activity is to provide idea of the reverse of
derivatives.
Directions: Complete the if-then statement below using rules in
derivatives.
1. If 𝑓(𝑥) = 4, then 𝑓′(𝑥) = ______________.
2. If 𝑔′(𝑥) = 3𝑥, then 𝑔(𝑥) = ______________. 3. If 𝑦 = 5𝑥2 − 3𝑥,

then ______________.
4. If 𝑦′ = 5, then 𝑦 = ______________.
5. If 𝑓(𝑥) = 2𝑥3 − 5𝑥2 − 3𝑥 then, 𝑓′′(𝑥) = ______________.
6. If ℎ′′(𝑥) = 20𝑥3 − 3𝑥, then ℎ′(𝑥) = ______________.

______________, then

, then 𝑓′(𝑥) = ______________.

, then 𝑦 = ______________.
, then 𝑡′(𝑥) = ______________
and 𝑡"(𝑥) = ______________.

CHAT TIME

In the previous lessons, we have shown the different ways for the finding the
derivatives of algebraic functions. Suppose the process is reversed, i.e., the
derivative or differential of a function is given, and we are asked to find the
function. The process of finding it is called antidifferentiation. An
antiderivative of a function 𝑓(𝑥) is a function 𝐹(𝑥).

Consider the function 𝑓(𝑥) = 𝑥5. Applying the power rule of differentiation, its
derivative is 5𝑥4 or in symbol 𝑓′(𝑥) = 5𝑥4. Suppose we reverse the operation

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Therefore, the antiderivative of 𝑓(𝑥) = 5𝑥4 is 𝐹(𝑥) = 𝑥5 + 𝐶 where 𝐶 represents

any real number.

Example 1. Find the antiderivative(s) of 𝑓(𝑥) = 2𝑥.

Solutions: Applying the power rule of differentiation

Therefore, the antiderivative of (𝑥) = 2𝑥 is 𝐹(𝑥) = 𝑥2 + 𝐶

where 𝐶 represents any real number.

Example 2. Find the antiderivative(s) of 𝑓(𝑥) = 7𝑥6 + 6𝑥.

Solutions: Applying the sum and power rule of differentiation

Therefore, the antiderivative of 𝑓(𝑥) = 7𝑥6 + 6𝑥 is

𝐹(𝑥) = 𝑥7 + 3𝑥2 + 𝐶
where 𝐶 represents any real number.

Example 3. Find the antiderivative(s) of

Solutions: Applying the negative exponent rule, we can rewrite the given as
𝑓(𝑥) = −8𝑥−9 and using the power rule of differentiation.

Therefore, the antiderivative of

where 𝐶 represents any real number.

The Indefinite Integral

Considering again example 1 above, (𝑥) = 2𝑥, we can say that its
antiderivative can be 𝐹(𝑥) = 𝑥2 + 4, is 𝐹(𝑥) = 𝑥2 + (−5) or is 𝐹(𝑥) = 𝑥2 + 1
since 𝐶 represents any real number. This family of all antiderivatives of the
function 𝑓 is called the indefinite integral.

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Since 𝐹(𝑥) is an antiderivative of 𝑓(𝑥), then we can say that 𝐹(𝑥) + 𝑐 is the
indefinite integral of 𝑓(𝑥) and is denoted by;
∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥) + 𝐶

where the arbitrary constant 𝐶 is called the constant of integration. Function

𝑓(𝑥)is called the integrand while the symbol ∫ is just an elongated 𝑆 meaning
sum and denotes the operation of integration.
The process of finding it is called integration. The mathematic meaning
of “to integrate” is “to find a function whose derivative is given”.

Note that the symbols ∫ … 𝑑𝑥 must always go together as the symbol

for the derivative.

Example 1.
Evaluate the integral ∫ 8𝑥7𝑑𝑥.
Solution:
The integrand is 𝑓(𝑥) = 8𝑥7. Applying the power rule of differentiation,

remember that .
Thus, 𝐹(𝑥) = 𝑥8 and ∫ 8𝑥7𝑑𝑥 = 𝑥8 + 𝐶.

Example 2.
Evaluate the integral ∫ 4𝑢3𝑑𝑢.
Solution:
The integrand is 𝑓(𝑢) = 4𝑢3. Applying the power rule of differentiation,

remember that .
Thus, 𝐹(𝑢) = 𝑢4 and ∫ 4𝑢3𝑑𝑢 = 𝑢4 + 𝐶.

Example 3.
Evaluate the integral ∫ 3𝑡2 − 2𝑡 𝑑𝑡.
Solution:
The integrand is 𝑓(𝑡) = 3𝑡2 − 2𝑡. Applying the difference and power rule of

differentiation, remember that .

Thus, 𝐹(𝑡) = 𝑡3 − 𝑡2 and ∫ 3𝑡2 − 2𝑡 𝑑𝑡 = 𝑡3 − 𝑡2 + 𝐶.
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ESSENTIAL NOTES

If 𝐹 ′ (𝑥 ) = 𝑓 (𝑥 ), then 𝐹 (𝑥 )is an antideravative of 𝑓 (𝑥 ).

If 𝐹 ′ (𝑥 ) = 𝑓 (𝑥 ), then ∫ 𝒇(𝒙)𝒅𝒙 = 𝑭(𝒙) + 𝑪, for any real number 𝐶 .

SHOW ME WHAT YOU’VE GOT

TIME TO PRACTICE
Description: This practice activity will enable you to evaluate integral of
functions.
Directions: For the given pair of functions 𝐹 and 𝑓, validate whether 𝐹 is an
antiderivaive of 𝑓. If not give its correct antiderivative.

1. 𝑓(𝑥) = 6; 𝐹(𝑥) = 6𝑥 + 3

2. 𝑓(𝑥) = 30𝑥9; 𝐹(𝑥) = 3𝑥10

3. 𝑓(𝑥) = 30𝑥9; 𝐹(𝑥) = 3𝑥10

10. 𝑓(𝑥) = 6𝑥2 − 8𝑥 − 5; 𝐹(𝑥) = 2𝑥3 − 4𝑥2 − 5𝑥 + 5

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Evaluate the following integral

1. ∫ 𝑑𝑥 6. ∫(𝑥 − 1)3𝑑𝑥

2. ∫ 2𝑥𝑑𝑥 7. ∫(4𝑥3 − 3𝑥2 − 4𝑥 − 2)𝑑𝑥

3. ∫(2𝑥2 + 3𝑥 − 4)𝑑𝑥 8. ∫ 2(5 − 𝑥)𝑑𝑥

4. ∫(𝑥1/2 − 𝑥−2 + 2) 𝑑𝑥 9. ∫ 2𝑥3 − 4𝑥 − 1/2
5. ∫(2𝑥 + 1)2 𝑑𝑥

THIS IS IT! GOTCHA!

As you will find in multivariable calculus, there is often a number of

solutions for any given problem.

- John Forbes Nash

Topic: APPLYING THE BASIC RULES IN EVALUATING THE

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INTEGRAL
Time Frame: 6 days

OBJECTIVE

Apply the basic rules in evaluating the integral.

LOOKING BACK

Activity 1
“The Reverse”
Directions: Find the derivative of the following to get the hidden message.
Write the word at the top of the correct answer on the answer box.

1. f(x) = 𝑥3 (Pray) 6. tan x (Great)

2. f(x) = 𝑥−3 (Is) 7. cot x (When)
3. f(x) = 𝑥5 + 5𝑥2 (Life) 8. csc x (Life)
4. f(x) = sin x (When) 9. sec x (Pray)
5. f(x) = (Rough) 10. cos x (Is)

Answer Box

___________ ___________ ___________ ___________ ___________

−3𝑥−4 1
cos x 5𝑥4 + 10𝑥 3𝑥2
2(√ 𝑥)

___________ ___________ ___________ ___________ ___________

-csc x cot x sec x tan x
−𝑐𝑠𝑐2𝑥 -sin x 𝑠𝑒𝑐2𝑥

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LET’S GET STARTED

Activity 2
“Search for Rule”
Directions: Complete the table by placing the correct given in each column.
(The students will be called randomly in which they will be given set of given
including the function and it’s integral. Let them place it on the correct rule of
integration applied.
Integration Rules FUNCTION INTEGRAL
Constant
Multiplication by Constant
Power Rule (𝑛 ≠ −1)
Sum Rule
Difference Rule
Integration by Parts
Substitution Rule

Given:

FUNCTION INTEGRAL FUNCTION INTEGRAL

ax + c
∫ 𝑎 𝑑𝑥 ∫(𝑓 − 𝑔)𝑑𝑥 ∫ 𝑓 𝑑𝑥 − ∫ 𝑔 𝑑𝑥

c∫ 𝑓 (𝑥)𝑑𝑥
∫ 𝑐𝑓 (𝑥)𝑑𝑥 ∫ 𝑢𝑣𝑑𝑥 𝑢 ∫ 𝑣𝑑𝑥 − ∫ 𝑢′(∫ 𝑣𝑑𝑥) 𝑑𝑥

∫ 𝑥𝑛 𝑑𝑥 𝑥𝑛+1 ∫ 𝑓(𝑔(𝑥))𝑔′(𝑥)𝑑𝑥 ∫ 𝑓(𝑢)𝑑𝑢

𝑛+1

∫(𝑓 + 𝑔)𝑑𝑥 ∫ 𝑓 𝑑𝑥 + ∫ 𝑔 𝑑𝑥

Guide Questions:
1. What made you decide to place the assigned given to you on its
corresponding rule?
2. What can you observe about the integral rules?
3. Is derivative connected to integral? Explain.

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CHAT TIME

Integration is a way of adding slices to find the whole.

Example: What is an integral of 2x?
We know that the derivative of x2 is 2x, so an
integral of 2x is x2.

The integral of many functions are well known, and there are useful
rules to work out the integral of more complicated functions. The following are
examples of how these rules were used.
1. Constant Rule
Example: What is ∫ 2𝑑𝑥?
Solution: Use the constant rule
∫ 𝑎𝑑𝑥 = 𝑎𝑥 + 𝑐
∫ 2𝑑𝑥 = 2𝑥 + 𝑐
2. Power Rule
Example: What is ∫ 𝑥3𝑑 ?
Solution: Use the power rule where n = 3

3. Multiplication by Constant
Example: What is ∫ 6𝑥2𝑑𝑥
Solution: 6 can be moved outside the integral:
∫ 6𝑥2𝑑𝑥 = 6 ∫ 𝑥2𝑑𝑥

Then, use the power rule on 𝑥2

Simplify:

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4. Sum Rule
Example: What is ∫ cos 𝑥 + 𝑥 𝑑𝑥?
Solution: Use the sum rule:
∫ cos 𝑥 + 𝑥𝑑𝑥 = ∫ cos 𝑥 𝑑𝑥 + ∫ 𝑥 𝑑𝑥

5. Difference Rule
Example: What is ∫ 𝑒𝑤 − 3𝑑𝑤?
Solution: Use the difference rule:
∫ 𝑒𝑤 − 3𝑑𝑤 = ∫ 𝑒𝑤𝑑𝑤 − ∫ 3𝑑𝑤
= 𝑒𝑤 − 3𝑤 + 𝑐
6. Integration by Parts
It is a special method of integration that is often useful when two functions are
multiplied together, but is also helpful in other ways.
See this rule:
∫ 𝑢𝑣𝑑𝑥 = 𝑢 ∫ 𝑣𝑑𝑥 − ∫ 𝑢′(∫ 𝑣𝑑𝑥)𝑑𝑥 Where: u is the
function u(x) v is the function v(x) In diagram:

Example: What is ∫ 𝑥𝑐𝑜𝑠 (𝑥)𝑑𝑥?

Solution:
Step 1: First choose which functions for u and v:
u=x
v = cosx
Step 2: Differentiate u: u’ = x’
=1
Step 3: Integrate v: ∫ 𝑣𝑑𝑥 = ∫ cos(𝑥) 𝑑𝑥 = 𝑠𝑖𝑛𝑥

Step 4: Put it together

Step 5: Simplify and solve
𝑥 𝑠𝑖𝑛𝑥 − ∫ 𝑠𝑖𝑛𝑥 𝑑𝑥

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𝑥 𝑠𝑖𝑛𝑥 + cos 𝑥 + 𝑐
Note: To remember ∫ 𝑢 𝑣 𝑑𝑥; use (u integral of v) minus integral of (derivative u, integral v)
7. Integration by Substitution
It is also called “u-substitution” or the “reverse chain rule”. It is a method to
find the integral, but only when it can be set up in a special way. The first and
vital step is to be able to write the integral in the form below:
Note: g(x) and its derivative g’(x).

Example:
∫ cos(𝑥2)2𝑥 𝑑𝑥; Step 1: Set up the
integral
f = cos and g=𝒙𝟐 and its derivative 2x Step 2:
Do the substitution
cos 𝑥2 = cos 𝑢
2𝑥 𝑑𝑥 = 𝑑𝑢
Step 3: Integrate
∫ cos(𝑢)𝑑𝑢 = sin(𝑢) + 𝑐
Step 4: Finally, put 𝑢 = 𝑥2 back again:
sin(𝑥2) + 𝑐
Therefore: ∫ 𝒄𝒐𝒔(𝒙𝟐)𝟐𝒙𝒅𝒙 = 𝒔𝒊𝒏 (𝒙𝟐) + 𝒄
Note: This rule only works on some integrals and somehow needs rearranging to apply.

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ESSENTIAL NOTES

THINGS TO REMEMBER…
• The process of finding the function whose derivative is given is
called antidifferentiation; it is the reverse of differentiation.
• If f’(x) = f(x), then f(x) is an antiderivative of f’(x).
Different Integration Rule
Rules Function Integral

Constant ∫ 𝑎 𝑑𝑥 ax + dx

Multiplication by ∫ 𝑐𝑓 (𝑥)𝑑𝑥 c∫ 𝑓 (𝑥)𝑑𝑥

Constant
Power Rule
∫ 𝑥𝑛 𝑑𝑥
Sum Rule
∫(𝑓 + 𝑔)𝑑𝑥 ∫ 𝑓 𝑑𝑥 + ∫ 𝑔 𝑑𝑥

Difference Rule
∫(𝑓 − 𝑔)𝑑𝑥 ∫ 𝑓 𝑑𝑥 − ∫ 𝑔 𝑑𝑥

Integration by Parts
∫ 𝑢𝑣𝑑𝑥 𝑢 ∫ 𝑣𝑑𝑥 − ∫ 𝑢′(∫ 𝑣𝑑𝑥) 𝑑𝑥

Integration by ∫ 𝑓(𝑔(𝑥))𝑔′(𝑥)𝑑𝑥 ∫ 𝑓(𝑢)𝑑𝑢

Substitution

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SHOW ME WHAT YOU’VE GOT

Activity 3
“Rule it”
Directions: Do what is instructed to solve the problem below through applying
the rules in integrals.

Problem: Philippines as one of the countries which is prone to typhoons

usually experienced floods in different areas. Morong, Rizal is one of the
affected areas when typhoons Inday and Josie hit our country this 2018.
This resulted to heavy flooding in different barangays in Morong particularly
along Baranggay San Juan. The DRRMO of Morong and Baranggay
officials work hand in hand to help the commuters pass on the affected
areas.
If Albert one of the stranded residents of Baranggay in Savemore Morong,

how he will reach his home safely?

To help Albert, solve the following integrals using the application of rules in

integrals. Show your complete solutions.

𝑧
What is ∫ 𝑑𝑥
(𝑧 +1)2
1. Help Albert to pass at
the area of Rizal Solution:
Provincial Hospital

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2. Help him ride on a

bicycle to cross the part
of Morong Cooperative

3. Help him pass the area of 2 4

What is ∫ 𝑏 −42𝑏 𝑑𝑥
Morong NHS which also has 𝑏
high water level dueSolution:

to heavy rain.
What is ∫ 8𝑔 + 4𝑔3 − 6𝑔2 𝑑𝑔. Apply the
sum and difference rule, multiplication by
constant and power rule to solve it.
Solutions:

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4. Help him ride on a Suppose the velocity of a car travelling in

boat to cross the area Morong town proper at time t is v(t) = 3t +5.
with the highest water At t=0, find the distance function s(t).
level caused by flood.
Solution:

The manufacturers of a certain type of

automobile advertise in Morong, Rizal
accelerate from 0 to 100 kph in 1 minute. If
its acceleration is constant, how far will the
car go in this length of time?
Solution:

5. Help him ride on a

bicycle to reach the safe
part for him to go home.

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THIS IS IT! GOTCHA!

“If we want to improve ourselves, learn how to integrate oneself to new

things without violating any rules”.
-Rhenelee S. Ramos-

Topic: Antiderivative of a Function

OBJECTIVE

Determine whether a function F is an antiderivative of a given function f.

LOOKING BACK

Time Frame: 2 Hours

Heal My Broken Heart
Description: This activity will test your memory on basic rules on
integration.
Directions: Heal the broken hearts by matching the function with its
corresponding integral.

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∫ (𝑓 + 𝑔) 𝑑𝑥
_____ 1. A 𝑎𝑥 + 𝑐

∫ 𝑐𝑓𝑛(𝑥 ) 𝑑𝑥 𝑥2
_____ 2. ∫ 𝑥 𝑑𝑥 B c∫ 𝑓 (+
𝑥 )𝑐𝑑𝑥
D 2
c

1
∫
∫ 𝑥𝑥𝑑𝑥𝑑𝑥 ∫ 𝑓𝑑𝑥 + 𝑐∫ 𝑔𝑑𝑥
_____ 3. C
E 𝑒𝑥 +

∫ (𝑓 − 𝑔) 𝑑𝑥 𝑥 𝑛 +1
_____ 4. F +𝑐
𝑛+1

∫ 𝑎 𝑑𝑥
_____ 5. G 𝑙𝑛|𝑥 | + 𝑐

∫ 𝑒 𝑥 𝑑𝑥 ∫ 𝑓𝑑𝑥 − ∫ 𝑔𝑑𝑥
H
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_____ 6.

_____ 7.

_____ 8

LET’S GET STARTED

A Look From The Past

Description: Discover the famous line of Benigno Aquino Jr. before

the Asia Society in New York on August 4, 1980.
Directions: Differentiate each functions to decode the answer.

3𝑥2 + 6𝑥 A. 𝐹(𝑥) = 𝑥3 + 3𝑥2 − 4 4𝑥3 − 3𝑥2

THE IS
2𝑥 + 2 3𝑥2 − 2𝑥
MANKIND
DYING

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3𝑥2 + 6𝑥 − 1 B. 𝐹(𝑥) = 𝑥3 + 𝑥2 − 1 2𝑥 − 2
GIVE FOR

4𝑥3 + 3𝑥2 C. 𝐹(𝑥) = 𝑥4 − 𝑥3 − 3 4𝑥3 + 3𝑥9

BEST LOVE
3𝑥2 + 2𝑥 D. 𝐹(𝑥) = 𝑥4 − 3𝑥3 + 8 4𝑥3 − 9𝑥2
FILIPINO WORTH
3𝑥3 − 2𝑥 E. 𝐹(𝑥) = 𝑥2 + 2𝑥 − 5 2𝑥2 − 2𝑥
PEOPLE
REVENGE
F. 𝐹(𝑥) = 𝑥2 − 2𝑥 + 7

A B C D E F

CHAT TIME
To determine whether a function F is an antiderivative of a given function f, simply
differentiate the given function F. The process of finding the function whose
derivative is given is called anti-differentiation, it is the inverse of differentiation.

Example A: Determining whether the function F is antiderivative of function f.

Function F Function f
1.
2.
3.
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1. 𝐹(𝑥) = 25𝑥
𝐹′(𝑥) = 25 hence, is an antiderivative of
.

2.
hence, is an antiderivative of
.

3.
hence, is antiderivative
of .

Example B: Finding antiderivative of .

4. Find an antiderivative of .
To find a function 𝐹 (𝑥 ) whose derivative is 8𝑥 5 , work backwards. We
have learned that the derivative of 𝑎𝑥 𝑛 is 𝑛𝑎 𝑥 𝑛 −1 .
If 𝑛𝑎 𝑥 𝑛 −1 is 8𝑥 5 then 𝑛 − 1 = 5, hence, 𝑛 = 6
8 8 4
Also, 𝑛𝑎 = 8, ℎ𝑒𝑛𝑐𝑒, 𝑎 = 𝑛 ; 𝑎 = 6 = 3
𝟒
Therefore, 𝟑 𝒙𝟔 is the antiderivative of 𝟖𝒙𝟓 .
4
We can check this by differentiating 3 𝑥 6 , and the answer will be 8𝑥 5 .

5. Find an antiderivative of
If 𝑛𝑎 𝑥 𝑛 −1 is −15𝑥 3 then 𝑛 − 1 = 3, hence, 𝑛 = 4
15 15
Also, 𝑛𝑎 = −15, ℎ𝑒𝑛𝑐𝑒, 𝑎 = − 𝑛
; 𝑎=− 4
15
Therefore, − 4
𝒙𝟒 is the antiderivative of −𝟏𝟓𝒙𝟑.
15
We can check this by differentiating − 4
𝑥 4 , and the answer will be

−15𝑥 3 .

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ESSENTIAL NOTES

Anti-differentiation of a Function
Anti-differentiation or integration is the reverse process of differentiation.
A function 𝐹 (𝑥 ) is called an antiderivative for the function 𝑓 (𝑥 ) if 𝐹 ′ (𝑥) =
𝑓 (𝑥 ).
To determine whether a function 𝐹 is an antiderivative of a given function 𝑓 ,
simply get the derivative of the function 𝐹 .

SHOW ME WHAT YOU’VE GOT

Oh Math! No!
Description: This activity will test your ability in determining the anti-derivative
of a function.
What do you call a specific developmental disability in learning
arithmetic?
Directions: Determine the anti-derivative of the following functions. Write the
corresponding letters of the correct answers on
the box.

1. U

2. Y
3. L 𝐹(𝑥) = 8 𝑙𝑛|𝑥|

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4. D 𝐹(𝑥) = 2𝑥3 − 2𝑥2 + 3𝑥

5. E 𝐹(𝑥) = 18𝑥2 − 12𝑥 + 8
3
6. C

7. X

8. S

I
A 𝐹(𝑥) = 2𝑥3

1 2 3 4 5 6 4 8 6 7 5

THIS IS IT! GOTCHA!

Our Creator designed us to be different from one another, but life becomes
more meaningful when we use our differences to integrate with each other.
- Cristy Caguntas-Cruz

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Topic: ANTIDERIVATIVE OF A FUNCTION

Time Frame: 5 days

OBJECTIVES

Calculate the specific antiderivative of a function.

LOOKING BACK

Activity 1: The Who?

Trivia Question: Who is the German philosopher, mathematician, and
political adviser, important both as a metaphysician and as a logician and
distinguished also for his independent invention of the differential and integral
calculus?

Directions: To answer the trivia question above,evaluate the antiderivatives of

the following and write the letter of your answer in the decoder.
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Answer Box

LET’S GET STARTED

Activity 2: Pair It!

Directions: Given the functions below,find the appropriate integral from the
answer box. Functions

_____________1.
_____________ 2.

_____________3.
′
_____________4.
_____________5.
_____________6.
_____________7.
_____________8.
_____________9.
_____________10.

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Answer Box
[𝑔(𝑥 )]𝑛 +1 +C
+ 𝐶, 𝑛 ≠ 1 C
𝑛+1

CHAT TIME

We have learned that Antidifferentiation is the process of finding

the set of all antiderivatives of a given function. This process is more
challenging than differentiation. Thus, our knowledge of the basic
antidifferentiation formulas is very important and need to be memorized
to be able to find the accurate antiderivates of a given function.

Consider the following:

A. Theorems on Integrals yielding the exponential and logarithmic functions.
1. ∫ 𝑒𝑥 𝑑𝑥 = 𝑒𝑥+ C

3.
B. Antiderivatives of Trigonometric Functions
1. ∫ sin 𝑥 𝑑𝑥 = −cos 𝑥 + 𝐶
2. ∫ cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝐶
3. ∫ 𝑠𝑒𝑐2 𝑥 𝑑 = tan𝑥 + 𝐶
4. ∫ 𝑐𝑠𝑐2𝑥 𝑑𝑥 = −cot 𝑥 + 𝐶
5. ∫ sec 𝑥 tan 𝑥 𝑑𝑥 = sec 𝑥 𝑑𝑥 + 𝐶
6. ∫ csc 𝑥 cot 𝑥 𝑑𝑥 = − csc 𝑥 𝑑𝑥

Examples:
Evaluate the following:
1. ∫ 𝑒4𝑥 𝑑𝑥

Solution.

; Let 𝑢 = 4𝑥.Then 𝑢 = 4𝑑𝑥.

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2. ∫ 35𝑥 𝑑𝑥

Solution.

Let

3.
Solution.

Let

4. ∫ cos(5𝑥 + 1)𝑑𝑥

Solution.

Let

5. ∫ 𝑒sin 𝑥 cos 𝑥 𝑑𝑥 Solution.

Let 𝑢 = sin 𝑥, cos 𝑥𝑑𝑥 = 𝑑𝑢. 𝑇ℎ𝑢𝑠,

∫ 𝑒sin 𝑥 cos 𝑥 𝑑𝑥 = ∫ 𝑒𝑢 𝑑𝑢 = 𝑒𝑢 + 𝐶 = 𝑒sin 𝑥 + 𝐶

ESSENTIAL NOTES

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THINGS TO REMEMBER…
To find the antiderivatives yielding:
A. The exponential and logarithmic functions, remember the
following:
1. ∫ 𝒆𝒙 𝒅𝒙 = 𝒆𝒙 + C

3.
B Trigonometric Functions
1. ∫ 𝐬𝐢𝐧 𝒙 𝒅𝒙 = −𝐜𝐨𝐬 𝒙 + 𝑪

2. ∫ 𝐜𝐨𝐬 𝒙 𝒅𝒙 = 𝐬𝐢𝐧 𝒙 + 𝑪

3. ∫ 𝒔𝒆𝒄𝟐 𝒙 𝒅 = 𝐭𝐚𝐧 𝒙 + 𝑪

4. ∫ 𝒄𝒔𝒄𝟐𝒙 𝒅𝒙 = −𝐜𝐨𝐭 𝒙 + 𝑪

5. ∫ 𝐬𝐞𝐜 𝒙 𝐭𝐚𝐧 𝒙 𝒅𝒙 = 𝐬𝐞𝐜 𝒙 𝒅𝒙 + 𝑪

6. ∫ 𝐜𝐬𝐜 𝒙 𝐜𝐨𝐭 𝒙 𝒅𝒙 = − 𝐜𝐬𝐜 𝒙 𝒅𝒙

SHOW ME WHAT YOU’VE GOT

Supreme Antiderivatives!

Find the antiderivative of the following .

THIS IS IT! GOTCHA!

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“Success is derived from hard work and patience after integrating oneself in doing
something that is worthwhile.”

-Marites G. Ancheta

29