# Introduction to integral calculus

```INTRODUCTION TO
INTEGRAL
CALCULUS
ENGR. REIMON MANERA
INTEGRATION
• Reverse process of differentiation
•The Antiderivative
• Can be Indefinite or Definite
Definite Integral
Indefinite Integral
• Unbounded Integrals
• Bounded Integrals
• Contains Arbitrary Constant (C)
• Does not contain Arbitrary Constants
• โซ ๐ ๐ญ = ๐๐ ๐ ๐ ืฌโฌ+ ๐ช
•
๐
โซ๐ ๐ืฌโฌ
Where:
โซืฌโฌ
= the integral sign
๐ ๐ ๐๐ = the integrand
x
= the variable of integration
๐ญ ๐ + ๐ช = indefinite integral, unknown
๐ช
= arbitrary constant
a,b
= lower and upper limit
๐
๐ ๐๐ = ๐ญ ๐ แฎ
๐
Basic Properties of Integration
1. โซ๐ ืฌโฌ
๐ ๐ฅ
=๐ ๐ฅ +๐ถ
a. โซ ๐ฅ = ๐ฅ๐ ืฌโฌ+ ๐ถ
b. โซ ๐ = ๐๐ ืฌโฌ+ ๐ถ
c. โซ ๐ฅ(๐ ืฌโฌ2 + 3๐ฅ) = ๐ฅ 2 + 3๐ฅ + ๐ถ
d. โซ = )๐ฅ(๐๐๐ ๐ ืฌโฌsin(๐ฅ) + ๐ถ
• ๐[โซ( ืฌโฌln ๐ฅ + ๐ฅ 2 )]
Basic Properties of Integration
2. Power Formula. โซืฌโฌ
b.
=
๐ฅ ๐+1
๐+1
+๐ถ
c.
a. โซ ๐ฅ ืฌโฌ3 ๐๐ฅ =
1 4
โซืฌโฌ−1 ๐ฅ ๐๐ฅ
๐ฅ ๐ ๐๐ฅ
=
๐๐ฅ
โซ๐ฅ ืฌโฌ
Basic Properties of Integration
3. โซ๐ฅ๐ ๐ฅ ๐ ืฌ ๐ = ๐ฅ๐)๐ฅ(๐๐ ืฌโฌ
a. โซ ืฌโฌ4๐๐ฅ =
b. โซ ืฌโฌ5๐ฆ 4 ๐๐ฆ =
c. โซ ืฌโฌ3 3 ๐ฅ ๐๐ฅ
Basic Properties of Integration
4. โซ )๐(๐[ืฌโฌ&plusmn; ๐(๐) &plusmn; โฏ ]๐๐ = โซ ๐๐ ๐ ๐ ืฌโฌ&plusmn; โซ ๐ ๐ ืฌโฌ&plusmn; โฏ
a. โซ ๐ฅ(ืฌโฌ3 + 2๐ฅ 2 − ๐ฅ + 3)๐๐ฅ =
b. โซ ๐ฅ(ืฌโฌ2 + 1)2 ๐๐ฅ =
Basic Properties of Integration
4. โซ )๐(๐[ืฌโฌ&plusmn; ๐(๐) &plusmn; โฏ ]๐๐ = โซ ๐๐ ๐ ๐ ืฌโฌ&plusmn; โซ ๐ ๐ ืฌโฌ&plusmn; โฏ
2
เถฑ (๐ฅ + 2)3 =
0
2
เถฑ (๐ฅ + 2)3 =
0
```