this about all the main ideas with examples of calculus 3 and the main rules and simple integrals

1. Math
Simple Integral
- Youssef Mansour
- Mohammad Mansour
- Jawad Ayoub
- Ali Aalawiye
Made by: Instructor: - Mohammad Al-
Samoury

2. Basic Approaches
•
csc 3𝑥 cot 3𝑥 𝑒csc 3𝑥
𝑒csc 3𝑥 𝑒csc 3𝑥 − 4
dx = -
1
3
𝑑𝑢
𝑢 𝑢2 − 𝑎2
= -
1
6
sec−1
( 𝑒csc 3𝑥
) + C
• 𝑢 = 𝑒csc 3𝑥
a = 2
• 𝑢′
= −3 csc 3𝑥 cot 3𝑥 𝑒csc 3𝑥
• −
𝑑𝑢
3
= csc 3𝑥 cot 3𝑥 𝑒csc 3𝑥

3. Integration By Part
• (𝑥3 + 2𝑥 − 1)𝑒2𝑥 𝑑𝑥 =
1
2
(𝑥3 + 2𝑥 − 1)𝑒2𝑥 −
1
4
(3𝑥2 + 2)𝑒2𝑥 +
1
8
(6𝑥)𝑒2𝑥 −
1
16
𝑒2𝑥 + 𝑐
• 𝑥3 + 2𝑥 − 1 𝑒2𝑥
• 3𝑥2 + 2
1
2
𝑒2𝑥
• 6𝑥
1
4
𝑒2𝑥
• 6
1
8
𝑒2𝑥
• 0
1
16
𝑒2𝑥
+
+
-
-

4. Integration By Part
• 𝑥 𝑠𝑒𝑐2 𝑥 𝑑𝑥 = 𝑢𝑣 − 𝑣 𝑑𝑣 = 𝑥 𝑡𝑎𝑛 − tan 𝑥 𝑑𝑥
• 𝑢 = 𝑥 𝑑𝑣 = 𝑠𝑒𝑐2 𝑥 𝑑𝑥
• 𝑑𝑢 = 𝑑𝑥 𝑣 = tan 𝑥
• tan 𝑥 𝑑𝑥 =
sin 𝑥
cos 𝑥
𝑑𝑥 = − ln cos 𝑥

5. Trigonometric Integral
• A)
• Case 1: 𝑠𝑖𝑛5 𝑥= sin 𝑥 (1−𝑐𝑜𝑠2 𝑥)
2
• Case 2: 𝑐𝑜𝑠5
𝑥= cos 𝑥 (1−𝑠𝑖𝑛2
𝑥)
2
• Case 3: 1. 𝑠𝑖𝑛22𝑎𝑥 = 1 −
cos 2𝑎𝑥
2
2. 𝑐𝑜𝑠2
2𝑎𝑥 = 1 +
cos 2𝑎𝑥
2• B)
• Case 1: 𝑠𝑒𝑐4
𝑥= (𝑡𝑎𝑛2
𝑥 + 1) 𝑠𝑒𝑐2
𝑥
• Case 2: 𝑠𝑒𝑐3 𝑥 𝑡𝑎𝑛3 𝑥 = 𝑠𝑒𝑐2 𝑥 (𝑠𝑒𝑐2 𝑥 - 1) 𝑠𝑒𝑐 𝑥 𝑡𝑎𝑛 𝑥
• Case 3: 𝑠𝑒𝑐3 𝑥 = 𝑠𝑒𝑐 𝑥 𝑠𝑒𝑐2 𝑥 = 𝑠𝑒𝑐 𝑥 (𝑠𝑒𝑐2 𝑥+1)
• Case 4: 𝑠𝑒𝑐 𝑥 𝑡𝑎𝑛2
𝑥= 𝑠𝑒𝑐 𝑥 (𝑠𝑒𝑐2
𝑥 - 1)

6. Trigonometric Integral
• C)
• 1 ± cos 2𝑎𝑥 = 2 𝑐𝑜𝑠2 𝑎𝑥 = 1 + cos 2 ax
= 2 𝑠𝑖𝑛2 𝑎𝑥 = 1 – cos 2 ax
• D)
• Case 1: {sin ax sin bx => sin ax sin bx = ½ [cos(a-b) x – cos (a +b)x]
• Case 2: {sin ax cos bx => sin ax cos bx = ½ [sin(a-b) x – sin (a +b)x]
• Case 3: {cos ax cos bx => cos ax cos bx = ½ [cos(a-b) x + cos (a +b)x]

7. Trigonometric Substitution
• 𝑎2 − 𝑢2 𝑑𝑢 𝑎 = ℎ2 − 𝑏2 𝑑𝑢
• 𝑢2 − 𝑎2 𝑑𝑢 𝑏 = ℎ2 − 𝑎2 𝑑𝑢
• 𝑎2 + 𝑢2 𝑑𝑢 ℎ = 𝑎2 − 𝑏2 𝑑𝑢
𝑏 𝑎
ℎ

8. Partial Fraction
• J =
11𝑥– 10
𝑥2−𝑥
𝑑𝑥 = 10 ln 𝑥 + ln 𝑥 − 1 + 𝐶
•
𝐴 + 𝐵 = 1 ⇒ 𝐵 = 1
−𝐴 = −10 ⇒ 𝐴 = 10

9. The End
Sir Isaac Newton