W E E K 1 : B A S I C I N T E G R AT I O N R U L E S

LEARNING OUTCOMES

At the end of the lesson, the students should be able to:

• Differentiate the indefinite and definite integrals;

• Evaluate simple integrals by reversing the process of differentiation

• Explain the need for a constant of integration when finding indefinite

integrals, and

• Apply the basic integration formulas in evaluating the indefinite and definite
integrals.
 INTEGRAL CALCULUS

A branch of mathematics concerned with the theory of and applications (the

determination of lengths, areas and volumes and in the solution of
differential equations) of integrals and integration (Merriam – Webster
dictionary)

Antiderivatives are also known as the Indefinite Integrals. They are the
reverse operation of differentiation wherein the original function being
solved for, given the derivative. A function F(x) is referred to as
antiderivative of f(x) on an interval I when F ′ x = f(x) on all x of the
interval.
 INTEGRAL CALCULUS

If F’(x) = f(x) or d(F(x)) = f(x)dx , then the anti-derivative of f(x) is defined as the
indefinite integral of f(x) and is mathematically expressed as
න𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥) + 𝑐

where: f(x) is the integrand

is the integral symbol
x is the variable of integration
dx is the differential of the variable
F(x) + c is the antiderivative
c is the constant of integration
 INTEGRAL CALCULUS

• Integration is the process or method of getting the antiderivative of a given function.

• Differentiation (the process of taking the derivative) and integration or anti – differentiation (the
process of finding the integral) are inverse processes. That means that one basically “undoes” the other.

Suppose you start with the function f x = x 3 .

Differentiating f(x) gives,

d 3
x = 3x 2
dx

Integrating this derivative, we get

x3
න 3x 2 = 3 + c = x3 + c
3

As a result, we get the original function back with the addition of the arbitrary constant of integration.
 T H E F U N DA M E N TA L T H E O R E M O F C A L C U L U S

If F(x) is the antiderivative of the function f(x) i.e. F ′ x = f(x), then

‫ ׬‬f x dx = ‫ ׬‬F"(x)dx = ‫ ׬‬dF = F x + C.

Thus, the integral of the differential of a function F is equal to the function itself plus an arbitrary
constant C.

Here are some examples

𝑥4 𝑥4
1. ‫ 𝑥 ׬‬3 𝑑𝑥 = ‫𝑑 ׬‬ 4
=
4
+𝐶

1 2𝑥 1
2. ‫ 𝑒 ׬‬2𝑥 𝑑𝑥 = ‫𝑑 ׬‬ 2
𝑒 = 𝑒 2𝑥 + 𝐶
2

1 1
3. ‫ ׬‬cos(5𝑥)𝑑𝑥 = ‫𝑑 ׬‬ 5
sin(5𝑥) = sin(5𝑥) + 𝐶
5
 2 TYPES OF INTEGRAL

Definite Integral - consists of the upper and lower limit, i.e. x = a and x = b
𝑏
න 𝑓(𝑥) 𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎)
𝑎

Indefinite Integral - does not have upper and lower limit

‫ )𝑥(𝐹 = 𝑥𝑑 )𝑥(𝑓 ׬‬+ 𝑐

 B A S I C I N T E G R AT I O N R U L E S

1. Identity Rule ‫ ׬‬du = u + c

2. Constant Rule න adu = a න du

n+1
u
3. Power Rule න un du = + c,n ≠ 1
n+1
4. Sum & Difference න f(x) ± g(x) dx = න f(x)dx ± න g(x)dx
du
5. Logarithmic Rule ‫׬‬ u
= Ln u + c , n ≠ 1 n ≥ 0
 EXAMPLES

Find the indicated integral.

1
Problem 1: ‫ ׬‬2𝑥 𝑑𝑥 3

Apply the following rules:

Power Rule , and

Constant Rule

Simplify the expression

Final Answer
 EXAMPLES

1
4
Problem 2 :‫(׬‬2x +
4 − 3x 4 )dx
x
Apply the following rules:

Power Rule , and

Constant Rule

Simplify the expression

Final Answer
 EXAMPLES

𝑦 3 −64
Problem 3 : ‫𝑦 ׬‬2 +4𝑦+16 𝑑𝑦
The first step is to rewrite the function and simplify it so we can apply the one of the basic
integration formulas.
Determine factors of the numerator in the integrand

(Recall: Difference of two perfect cubes)

Cancel out common factor

Integrate applying power rule and constant rule

Final Answer
 EXAMPLES

y3 −2y2 +y−1
Problem 4 :
‫׬‬ y2
dy

Write as separate fractions

Integrate each term separately applying the following rules:

Power Rule , and

Constant Rule

Logarithmic Rule

Simplify the expression

Final Answer
 EXAMPLES

Problem 5 :‫ ׬‬3𝑥 2 − 5𝑥 + 3 𝑑𝑥

Express as

Express the radicals into fractional exponent

Integrate each term separately and simplify

Final Answer

Note: To rewrite a radical using a fractional exponent, the power to which the
radicand is raised becomes the numerator and the root becomes the denominator.
 EXAMPLES

Problem 6 :‫ ׬‬x+4
x
dx

Write as separate fractions

Express the radicals into fractional exponents

Integrate each term separately

Simplify the expression

Final Answer
 EXAMPLES

2x2 +4x−3
Problem 7 :‫׬‬ x2
dx
Write as separate fractions

Integrate each term separately

Simplify the expression

Final Answer
 EXAMPLES

2
Problem 8 : ‫׬‬0 6𝑥 2 − 4𝑥 + 5 𝑑𝑥
Integrate each term separately

Recall:

Evaluate by replacing x by the given

values of the upper limit and lower limit

Simplify

Final Answer
 EXAMPLES
1
Problem 9 : ‫׬‬0 4 + 3 𝑥 − 2𝑥 𝑥 𝑑𝑥
Express radicals to fractional exponents

Integrate each term separately

Recall:

Evaluate by replacing x by the given values of the

upper limit and lower limit

Simplify the expression

Final Answer
 EXAMPLES
2
Problem 10 : ‫׬‬−2 2x − 1 3x + 4 dx
Multiply using FOIL method

Integrate each term separately

Simplify the expression

Recall:

Evaluate by replacing x by the given values of

the upper limit and lower limit

Simplify the expression

Final Answer
 EXAMPLES
4
Problem 11: ‫׬‬1 x x − 1 dx
Distribute x then rewrite radicals to fractional exponents

Integrate each term separately

Simplify the expression

Recall:

Evaluate by replacing x by the given values of the upper limit and

lower limit
 EXAMPLES

Problem 11:
2 1 2 1
= (32) − (16) − (1) − (1) Simplify
5 2 5 2

64 2 1
= −8 − −
5 5 2

49
= Final Answer
10
 REFERENCES
Printed References
1. Stewart, J. (2019). Calculus concepts and contexts 4th edition enhanced edition. USA :
Cengage Learning, Inc.
2. Canlapan, R.B.(2017). Diwa senior high school series: basic calculus. DIWA Learning
Systems Inc.
3. Hass, J., Weir, M. & Thomas, G. (2016). University calculus early transcendentals,global
edition. Pearson Education, Inc.Larson,
4. Bacani, J.B.,et.al. (2016). Basic calculus for senior high school. Book AtBP. Publishing
Corp.
5. Balmaceda, J.M. (2016). Teaching guide in basic calculus. Commission on Higher
Education.
6. Pelias, J.G. (2016). Basic calculus. Rex Bookstore.
7. Molina, M.F. (2016). Integral calculus. Unlimited Books Library Services & Publishing
Inc.
8. Larson, R & Edwards, B. (2012). Calculus. Pasig City, Philippines. Cengage Learning
Asia Pte Ltd

Electronic Resources
1. Pulham , John. Differential Calculus. Retrieved August 7, 2020 from http://
www.maths.abdn.ac.uk/igc/tch/math1002/dif
2. Weistein, Eric. Differential Calculus. Retrieved August 7, 2020 from
http://mathworld.wolfram.com/differential calculus
3. Thomas , Christopher . Introduction to Differential Calculus. Retrieved August 7, 2020
from http:// www.usyd.edu.aav/stuserv/document/maths_learning-center
4. Purplemath. Retrieved August 7, 2020 from http://www.purplemath.com/modules/