sheesh

1. ANTIDERIVATIVE OF
EXPONENTIAL
FUNCTION
WEEK 04

2. DIFFERENTIATION FORMULAS
• 𝑫𝒙 𝒆𝒙
= 𝒆𝒙
• 𝑫𝒙 𝒂𝒙
= 𝒂𝒙
𝐥𝐧𝐚
• 𝑫𝒙 𝐥𝐧𝐱 =
𝟏
𝒙

3. THEOREMS ON INTEGRALS YIELDING
THE EXPONENTIAL FUNCTION
•1. ∫ 𝒆𝒙
𝒅𝒙 = 𝒆𝒙
+ 𝒄
•2. ∫ 𝒂𝒙
𝒅𝒙 =
𝒂𝒙
𝒍𝒏𝒂
+ 𝐜 𝐡𝐞𝐧𝐜𝐞, 𝒂 > 𝟎 𝒘𝒊𝒕𝒉 𝒂 ≠
𝟏
•3. ∫ 𝒙−𝟏
𝒅𝒙 = ∫
𝟏
𝒙
𝒅𝒙 = 𝒍𝒏 𝒙 + 𝒄

4. THEOREMS ON INTEGRALS YIELDING
THE EXPONENTIAL FUNCTION
Example 1.
∫ 𝟑𝒆𝒙
𝒅𝒙
= 𝟑 ∫ 𝒆𝒙
𝒅𝒙
= 𝟑∫ 𝒆𝒙
+ 𝒄
Example 2.
∫
𝟏
𝟑
𝒆𝒙
𝒅𝒙
=
𝟏
𝟑
∫𝒆𝒙
𝒅𝒙
=
𝟏
𝟑
𝒆𝒙
𝒅𝒙
1. ∫ 𝒆𝒙
𝒅𝒙 = 𝒆𝒙
+
𝒄

5. THEOREMS ON INTEGRALS YIELDING
THE EXPONENTIAL FUNCTION
Example 1.
∫ 𝟑𝒙𝒅𝒙
=
𝟑𝒙
𝒍𝒏𝟑
+ 𝒄
2. ∫ 𝒂𝒙
𝒅𝒙 =
𝒂𝒙
𝒍𝒏𝒂
+ 𝐜
Example 2.
∫
𝟑𝒙+𝟏𝒅𝐱
= ∫(𝟐𝒙
) 𝟐𝟏
= 𝟐∫(𝟐𝒙
)𝒅𝒙
= 𝟐
𝟐𝒙
𝒍𝒏𝟐
+ 𝒄
Example 3
∫ (𝒆𝒙+𝟐𝒙)𝒅𝐱
= ∫𝒆𝒙𝒅𝒙 + ∫𝟐𝒙𝒅𝒙
= 𝒆𝒙
𝟐𝒙
𝒍𝒏𝟐
+ 𝒄

6. THEOREMS ON INTEGRALS YIELDING
THE EXPONENTIAL FUNCTION
Example 1.
∫
𝟐
𝒙
𝒅𝒙 = 𝟐
= 2∫
𝟏
𝒙
𝒅𝒙 +
𝒄
= 𝟐𝒍𝒏 𝒙 + 𝒄
3. ∫ 𝒙−𝟏
𝒅𝒙 = ∫
𝟏
𝒙
𝒅𝒙 = 𝒍𝒏 𝒙 + 𝒄
Example 2.
∫ 𝟑𝒙−𝟏𝒅𝐱 = 𝟑
= 3∫
𝟏
𝒙
𝒅𝒙 + 𝒄
= 𝟐∫(𝟐𝒙
)𝒅𝒙
= 𝟑𝒍𝒏 𝒙 + 𝒄