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
• ‫ 𝒙 𝑭 = 𝒙𝒅 𝒙 𝒇 ׬‬+ 𝑪
•
𝒃
‫𝒇 𝒂׬‬
Where:
‫׬‬
= the integral sign
𝒇 𝒙 𝒅𝒙 = the integrand
x
= the variable of integration
𝑭 𝒙 + 𝑪 = indefinite integral, unknown
𝑪
= arbitrary constant
a,b
= lower and upper limit
𝒃
𝒙 𝒅𝒙 = 𝑭 𝒙 ቮ
𝒂
Basic Properties of Integration
1. ‫𝑑 ׬‬
𝑓 𝑥
=𝑓 𝑥 +𝐶
a. ‫ 𝑥 = 𝑥𝑑 ׬‬+ 𝐶
b. ‫ 𝜃 = 𝜃𝑑 ׬‬+ 𝐶
c. ‫ 𝑥(𝑑 ׬‬2 + 3𝑥) = 𝑥 2 + 3𝑥 + 𝐶
d. ‫ = )𝑥(𝑛𝑖𝑠𝑑 ׬‬sin(𝑥) + 𝐶
• 𝑑[‫( ׬‬ln 𝑥 + 𝑥 2 )]
Basic Properties of Integration
2. Power Formula. ‫׬‬
b.
=
𝑥 𝑛+1
𝑛+1
+𝐶
c.
a. ‫ 𝑥 ׬‬3 𝑑𝑥 =
1 4
‫׬‬−1 𝑥 𝑑𝑥
𝑥 𝑛 𝑑𝑥
=
𝑑𝑥
‫𝑥 ׬‬
Basic Properties of Integration
3. ‫𝑥𝑑 𝑥 𝑓 ׬ 𝑘 = 𝑥𝑑)𝑥(𝑓𝑘 ׬‬
a. ‫ ׬‬4𝑑𝑥 =
b. ‫ ׬‬5𝑦 4 𝑑𝑦 =
c. ‫ ׬‬3 3 𝑥 𝑑𝑥
Basic Properties of Integration
4. ‫ )𝒙(𝒇[׬‬± 𝒈(𝒙) ± ⋯ ]𝒅𝒙 = ‫ 𝒙𝒅 𝒙 𝒇 ׬‬± ‫ 𝒙 𝒈 ׬‬± ⋯
a. ‫ 𝑥(׬‬3 + 2𝑥 2 − 𝑥 + 3)𝑑𝑥 =
b. ‫ 𝑥(׬‬2 + 1)2 𝑑𝑥 =
Basic Properties of Integration
4. ‫ )𝒙(𝒇[׬‬± 𝒈(𝒙) ± ⋯ ]𝒅𝒙 = ‫ 𝒙𝒅 𝒙 𝒇 ׬‬± ‫ 𝒙 𝒈 ׬‬± ⋯
2
න (𝑥 + 2)3 =
0
2
න (𝑥 + 2)3 =
0