The document discusses antidifferentiation, which is the process of reversing differentiation. It provides examples of differentiating functions to arrive at their derivatives, and then taking the derivatives and integrating them to arrive back at the original functions. The key points are: - Antidifferentiation/indefinite integration undoes the process of differentiation - If the derivative of f(x) is g(x), then an antiderivative of g(x) is f(x) - Several integration formulas and examples are provided to demonstrate how to find antiderivatives of common functions like polynomials, trigonometric, exponential, and logarithmic functions

1. Just like any other mathematical operation, the process of differentiation can be reversed. For example,
when we perform the differentiation of 𝑓(𝑥) = 𝑥3
.
𝒇(𝒙) = 𝒙𝟑
𝒇′(𝒙) = 𝟑𝒙𝟑−𝟏
𝒇′(𝒙) = 𝟑𝒙𝟐
Now if you begin with the function 𝒇(𝒙) = 𝟑𝒙𝟐
, reversing the process should yield the possible functions
below:
𝒇(𝒙) = 𝒙𝟑
𝒇(𝒙) = 𝒙𝟑
+ 𝟏
𝒇(𝒙) = 𝒙𝟑
− 𝟏
The reversing of the operation of differentiation is known as ANTIDIFFERENETIATION or INDEFINITE
INTEGRATION. If the derivative of 𝑓(𝑥) = 𝑥3
is 𝑓′(𝑥) = 3𝑥2
, then we say that an antiderivative of 𝑓(𝑥) =
3𝑥2
is 𝑓(𝑥) = 𝑥3
.
𝒇(𝒙) = 𝒙𝟑
𝒇′(𝒙) = 𝟑𝒙𝟐
∫ 𝒇′(𝒙) 𝒅𝒙 = 𝒇(𝒙) + 𝑪
FINDING THE ANTIDERIVATIVE OF A FUNCTION
BASIC INTEGRATION FORMULAS
Differentiation Formulas Integration Formulas
𝒅
𝒅𝒙
(𝑪) = 𝟎 ∫ 𝟎 𝒅𝒙 = 𝑪
𝒅
𝒅𝒙
(𝒌𝒙) = 𝒌 ∫ 𝒌 𝒅𝒙 = 𝒌𝒙 + 𝑪
𝒅
𝒅𝒙
(𝒌𝑭(𝒙)) = 𝒌𝑭′(𝒙) ∫ 𝒌 𝒇(𝒙) 𝒅𝒙 = 𝒌 ∫ 𝒇( 𝒙) 𝒅𝒙
𝒅
𝒅𝒙
(𝑭(𝒙) + 𝑮(𝒙)) = 𝑭′(𝒙) + 𝑮′(𝒙) ∫[𝒇(𝒙) + 𝒈(𝒙)]𝒅𝒙 = ∫ 𝒇(𝒙) 𝒅𝒙 + ∫ 𝒈(𝒙) 𝒅𝒙
DERIVATIVE
INTEGRAL

2. 𝒅
𝒅𝒙
(𝒙𝒏) = 𝒏𝒙𝒏−𝟏
∫ 𝒙𝒏
𝒅𝒙 =
𝒙𝒏+𝟏
𝒏 + 𝟏
+ 𝑪; 𝒏 ≠ −𝟏
EXAMPLE 1: Find the ∫ 2
SOLUTION:
∫ 𝟐 = 𝟐𝒙 + 𝒄
EXAMPLE 2: Find the ∫(𝑥 − 3)(𝑥 + 4)
SOLUTION:
∫(𝒙 − 𝟑)(𝒙 + 𝟒) = (𝒙 − 𝟑)(𝒙 + 𝟒)
= (𝒙𝟐
+ 𝒙 − 𝟏𝟐)
=
𝒙𝟐+𝟏
𝟐+𝟏
+
𝒙𝟏+𝟏
𝟏+𝟏
− 𝟏𝟐𝒙 + 𝑪
=
𝒙𝟑
𝟑
+
𝒙𝟐
𝟐
− 𝟏𝟐𝒙 + 𝑪
EXAMPLE 3: Find the ∫ √𝑥(2𝑥2
− 3𝑥 + 1)
SOLUTION:
∫ √𝑥(2𝑥2
− 3𝑥 + 1) = (𝒙
𝟏
𝟐)(2𝑥2
− 3𝑥 + 1)
= 𝟐𝒙
𝟏
𝟐
+𝟐
− 𝟑𝒙
𝟏
𝟐
+𝟏
+ 𝒙
𝟏
𝟐
= 𝟐𝒙
𝟓
𝟐 − 𝟑𝒙
𝟑
𝟐 + 𝒙
𝟏
𝟐
=
𝟐𝒙
𝟓
𝟐
+𝟏
𝟓
𝟐
+𝟏
−
𝟑𝒙
𝟑
𝟐
+𝟏
𝟑
𝟐
+𝟏
+
𝒙
𝟏
𝟐
+𝟏
𝟏
𝟐
+𝟏
+ 𝑪
=
𝟐𝒙
𝟕
𝟐
𝟕
𝟐
−
𝟑𝒙
𝟓
𝟐
𝟓
𝟐
+
𝒙
𝟑
𝟐
𝟑
𝟐
+ 𝑪
=
𝟒𝒙
𝟕
𝟐
𝟕
−
𝟔𝒙
𝟓
𝟐
𝟓
+
𝟐𝒙
𝟑
𝟐
𝟑
+ 𝑪

3. =
𝟒
𝟕
√𝒙𝟕 −
𝟔
𝟓
√𝒙𝟓 +
𝟐
𝟑
√𝒙𝟑 + 𝑪
TRIGONOMETRIC FUNCTIONS INTEGRATION FORMULAS
Differentiation Formulas Integration Formulas
𝒅
𝒅𝒙
(𝒔𝒊𝒏 𝒙) = 𝒄𝒐𝒔 𝒙 ∫ 𝒄𝒐𝒔 𝒙 𝒅𝒙 = 𝒔𝒊𝒏 𝒙 + 𝑪
𝒅
𝒅𝒙
(𝒄𝒐𝒔 𝒙) = −𝒔𝒊𝒏 𝒙 ∫ 𝒔𝒊𝒏 𝒙 𝒅𝒙 = −𝒄𝒐𝒔 𝒙 + 𝑪
𝒅
𝒅𝒙
(𝒕𝒂𝒏 𝒙) = 𝒔𝒆𝒄𝟐
𝒙 ∫ 𝒔𝒆𝒄𝟐
𝒙 𝒅𝒙 = 𝒕𝒂𝒏 𝒙 + 𝑪
𝒅
𝒅𝒙
(𝒄𝒐𝒕 𝒙) = −𝒄𝒔𝒄𝟐
𝒙 ∫ 𝒄𝒔𝒄𝟐
𝒙 𝒅𝒙 = −𝒄𝒐𝒕 𝒙 + 𝑪
𝒅
𝒅𝒙
(𝒔𝒆𝒄 𝒙) = 𝒔𝒆𝒄 𝒙 𝒕𝒂𝒏 𝒙 ∫ 𝒔𝒆𝒄 𝒙 𝒕𝒂𝒏 𝒙 𝒅𝒙 = 𝒔𝒆𝒄 𝒙 + 𝑪
𝒅
𝒅𝒙
(𝒄𝒔𝒄 𝒙) = −𝒄𝒔𝒄 𝒙 𝒄𝒐𝒕 𝒙 ∫ 𝒄𝒔𝒄 𝒙 𝒄𝒐𝒕 𝒙 𝒅𝒙 = −𝒄𝒔𝒄 𝒙 + 𝑪
EXAMPLE 4: Find the ∫(4 cos 𝑥 − 3 sin𝑥) 𝑑𝑥
SOLUTION:
∫(𝟒 𝐜𝐨𝐬 𝒙 − 𝟑 𝐬𝐢𝐧 𝒙) 𝒅𝒙 = 𝟒 ∫ 𝐜𝐨𝐬 𝒙 𝒅𝒙 − 𝟑 ∫ 𝐬𝐢𝐧 𝒙 𝒅𝒙
= 𝟒𝒔𝒊𝒏 𝒙 − 𝟑(−𝒄𝒐𝒔 𝒙) + 𝑪
= 𝟒𝒔𝒊𝒏 𝒙 + 𝟑𝒄𝒐𝒔 𝒙 + 𝑪
EXAMPLE 5: Find the ∫(𝑠𝑒𝑐2
𝑥 + 𝑐𝑠𝑐2
𝑥) 𝑑𝑥
SOLUTION:
∫(𝒔𝒆𝒄𝟐
𝒙 + 𝒄𝒔𝒄𝟐
𝒙) 𝒅𝒙 = 𝒕𝒂𝒏 𝒙 − 𝒄𝒐𝒕 𝒙 + 𝑪
EXPONENTIAL & LOGARITHMIC FUNCTIONS INTEGRATION FORMULAS
Differentiation Formulas Integration Formulas
𝒅
𝒅𝒙
(𝒆𝒙
) = 𝒆𝒙 ∫ 𝒆𝒙
𝒅𝒙 = 𝒆𝒙
+ 𝑪

4. 𝒅
𝒅𝒙
(𝒂𝒙) = 𝒂𝒙
𝒍𝒏 𝒂, 𝒂 > 𝟎 ∫ 𝒂𝒙
𝒅𝒙 =
𝒂𝒙
𝒍𝒏 𝒂
+ 𝑪, 𝒂 > 𝟎
𝒅
𝒅𝒙
(𝒍𝒏 𝒙) =
𝟏
𝒙
∫
𝒅𝒙
𝒙
= 𝒍𝒏 |𝒙| + 𝑪
EXAMPLE 6: Find the ∫(2𝑥
− 3𝑥
) 𝑑𝑥
SOLUTION:
∫(𝟐𝒙
− 𝟑𝒙) =
𝟐𝒙
𝒍𝒏 𝟐
−
𝟑𝒙
𝒍𝒏 𝟑
+ 𝑪
EXAMPLE 7: Find the ∫(
2
𝑥
− 3𝑒3
) 𝑑𝑥
SOLUTION:
∫ (
𝟐
𝒙
− 𝟑𝒆𝒙
) 𝒅𝒙 = 𝟐 ∫
𝒅𝒙
𝒙
− 𝟑 ∫ 𝒆𝟑
= 𝟐 𝒍𝒏 |𝒙| − 𝟑𝒆𝟑