Maths-Methods-Club-Integration-Cheat-Sheet-Sample - Anti-Differentiation & Definite Integrals Rules

Q: Why must the constant of integration be included after integrating?

The notes highlight that forgetting to add the constant leads to incomplete antiderivatives. Add “+ c” after integrating to capture all possible solutions.

Q: When do the antidifferentiation rules apply?

They apply only when the expression inside the brackets is linear. For other kinds, expand if possible before integrating or use a CAS calculator.

Q: How is a definite integral computed using an antiderivative?

If the function is continuous on [a, b], the notes state that the definite integral equals the antiderivative evaluated at b minus its value at a, i.e., F(b) − F(a).

Oct 13, 2026

Upload by Sophia Brooks

Integration notes covering key properties of integrals, including linearity with plus/minus splits and how constants affect antiderivatives. Includes a dedicated section on forgetting the constant, with reminders about adding “+ c” after integrating and how expressions in brackets must be linear for the antidifferentiation rules to apply. Summarizes several anti-differentiation patterns for powers, logarithms, and trigonometric functions, then consolidates the definite integral rule using an antiderivative and fundamental integral properties.

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Integration

Properties of Integrals

Forgetting to include the constant

Rules for Anti-Differentiation

Definite Integral via an Antiderivative

Integration |
Properties of Integrals

• ∫ 􀝂 ± 􀝃 (􀝔) 􀝀􀝔 = ∫ 􀝂 􀝀􀝔 ± ∫ 􀝃 􀝀􀝔
• ∫ 􀝇􀝂 􀝀􀝔 = 􀝇 ∫ 􀝂 􀝀􀝔, where 􀝇 is a real number. | | Forgetting to include the constant | | | --- | --- | --- | --- | | | | ‘+􀜿’after integrating or
including‘􀝀􀝔’when
integrating. | | | | | | | | | Note: The antidifferentiation rules | | | | | described ONLY apply if the expression inside the brackets
is linear.
If the expression is of any other kind, you may need to expand it if possible before integrating, or you must use your CAS calculator to integrate the function. | | | |
Rules for Anti-Differentiation

• ∫ (􀜽􀝔 + 􀜾)􀯡 􀝀􀝔 = (􀯔􀯫􀯔)􀳙11 + c, where 􀝊 ≠ −1
• ∫ 1􀯫 􀝀􀝔 = ∫ 􀝔 −1 􀝀􀝔 = log􀯘 (􀝔 ) + 􀜿
• ∫ ~~ ~~(􀯔􀯫􀯕) 􀝀􀝔 = ∫ (􀜽􀝔 + 􀜾)−1 􀝀􀝔 = 1􀯔 log 􀯘 (􀜽􀝔 + 􀜾) + 􀜿
• ∫ 􀝁 􀯫 􀝀􀝔 = 􀝁 􀯫 + 􀜿
• ∫ 􀝁 􀯞􀯫 􀝀􀝔 = 1􀯞 􀝁 􀯞􀯫 + 􀜿
• ∫ sin(􀜽􀝔) 􀝀􀝔 = cos 􀜽􀝔 + 􀜿
• ∫ sin(􀜽􀝔 + 􀜾) 􀝀􀝔 = − ~~ ~~1􀯔 cos(􀜽􀝔 + 􀜾) + 􀜿
• ∫ cos(􀜽􀝔) 􀝀􀝔 = 1􀯔 sin 􀜽􀝔 + 􀜿
• ∫ cos(􀜽􀝔 + 􀜾) 􀝀􀝔 = 1􀯔 sin 􀜽􀝔 + 􀜾 + 􀜿

| | | | | | Find the antiderivative of 3
􀯫
∫~~ ~~3􀝔 − ~~ ~~ 􀝀􀝔= ∫ 3􀝔1 − ~~ ~~~~ ~~ 􀝀􀝔

1
− ~~ ~~

2􀯫−2

3
= ~~ ~~
12
1 1
􀝔 2 − 2 􀝈􀝋􀝃􀯘 2􀝔 − 2 + 􀜿

1 1
= 6􀝔2 − 2 􀝈􀝋􀝃􀯘 2􀝔 − 2 + 􀜿= 6 − ~~ ~~12 􀝈􀝋􀝃􀯘 + 􀜿
| | | |
If 􀝂 is a continuous function on the interval [a, b] then
􀯕
∫ 􀝂 (􀝔) 􀝀􀝔 = [􀜨 ]􀯕􀯔􀯔
= 􀜨 􀜾 − 􀜨 􀜽
Where 􀜨 is the antiderivative of 􀝂 .
| |
• ∫􀯔􀯔 􀝂 􀝀􀝔 = 0
• ∫􀯕􀯔 􀝂 (􀝔) 􀝀􀝔 = − ∫􀯕􀯔 􀝂 􀝀􀝔
• ∫􀯕􀯔 􀝇􀝂 􀝀􀝔 = 􀝇 ∫􀯕􀯔 􀝂 (􀝔 ) 􀝀􀝔
• ∫􀯕􀯔 (􀝂 ± 􀝃 ) 􀝀􀝔 = ∫􀯕􀯔 􀝂 􀝀􀝔 + ∫􀯕􀯔 􀝃 (􀝔 ) 􀝀􀝔
• ∫􀯕􀯔 􀝂 (􀝔) 􀝀􀝔 = ∫􀯖􀯔 􀝂 (􀝔) 􀝀􀝔 + ∫􀯖􀯕 􀝂 (􀝔) 􀝀􀝔, providing
􀜽 < 􀜿 < 􀜾 | | --- | --- | --- | [www. Maths](www. MathsMethodsClub.com.au)[Methods](www. MathsMethodsClub.com.au)[Club.com.au](www. MathsMethodsClub.com.au) 24 [Facebook.com/Maths](Facebook.com/MathsMethodsClub)[Methods](Facebook.com/MathsMethodsClub)[Club](Facebook.com/MathsMethodsClub)

FAQs

Why must the constant of integration be included after integrating?

The notes highlight that forgetting to add the constant leads to incomplete antiderivatives. Add “+ c” after integrating to capture all possible solutions.

When do the antidifferentiation rules apply?

They apply only when the expression inside the brackets is linear. For other kinds, expand if possible before integrating or use a CAS calculator.

How is a definite integral computed using an antiderivative?

If the function is continuous on [a, b], the notes state that the definite integral equals the antiderivative evaluated at b minus its value at a, i.e., F(b) − F(a).

Oct 13, 2026

Upload by Sophia Brooks