📐 Concept diagram

05-01 - Antiderivatives

Phase: 5 | Subject: 05-01 Prerequisites: 04-04-differentiation-rules.md (power rule, derivatives of elementary functions) Next subject: 05-02-the-definite-integral.md

Learning Objectives

By the end of this subject, you will be able to:

1. Define antiderivatives and understand the relationship to derivatives
2. Use the power rule for integration
3. Integrate polynomials
4. Apply basic integration rules for elementary functions
5. Understand the constant of integration

Core Content

What is an Antiderivative?

An antiderivative of f(x) is a function F(x) such that F'(x) = f(x).

Example: f(x) = 2x An antiderivative is F(x) = x² because d/dx[x²] = 2x.

But x² + 5, x² - 3, and x² + C (for any constant C) are also antiderivatives!

General form: F(x) + C, where C is any constant.

The Indefinite Integral

The indefinite integral is the set of ALL antiderivatives:

$∫f(x)dx = F(x) + C
$

Key notation: - ∫ is the integral sign (elongated S for "sum") - f(x) is the integrand - dx means "with respect to x" - C is the constant of integration

Basic Integration Rules

Power Rule for Integration

∫xⁿdx = x^(n+1)/(n+1) + C   (for n ≠ -1)

Example: ∫x³dx = x⁴/4 + C Example: ∫x²dx = x³/3 + C Example: ∫x dx = x²/2 + C Example: ∫1 dx = x + C

Special Case: n = -1

∫(1/x)dx = ln|x| + C

This is because d/dx[ln(x)] = 1/x.

Constant Multiple Rule

∫k·f(x)dx = k·∫f(x)dx

Example: ∫3x²dx = 3∫x²dx = 3(x³/3) = x³ + C

Sum/Difference Rule

∫[f(x) ± g(x)]dx = ∫f(x)dx ± ∫g(x)dx

Example: ∫(3x² - 2x + 1)dx = x³ - x² + x + C

Integration of Elementary Functions

$∫eˣdx = eˣ + C
∫aˣdx = aˣ/ln(a) + C
∫sin(x)dx = -cos(x) + C
∫cos(x)dx = sin(x) + C
∫sec²(x)dx = tan(x) + C
∫csc²(x)dx = -cot(x) + C
∫sec(x)tan(x)dx = sec(x) + C
∫csc(x)cot(x)dx = -csc(x) + C
$

Verification: Differentiate the result to check.

Key Terms

• 05 01 Antiderivatives
• Basic Integration Rules
• Constant Multiple Rule
• Correct: A)
• Correct: B)
• Example 1: Polynomial
• Example 2: Exponential and trig
• Example 3: With constant multiple
• **Indefinite Integral

The indefinite integral - Integration of Elementary Functions - Power Rule for Integration - Special Case: n = -1**

Worked Examples

Example 1: Polynomial

∫(4x³ - 2x² + 5x - 7)dx = 4(x⁴/4) - 2(x³/3) + 5(x²/2) - 7x + C = x⁴ - (2/3)x³ + (5/2)x² - 7x + C

Example 2: Exponential and trig

∫(eˣ + sin(x) + 3)dx = eˣ - cos(x) + 3x + C

Example 3: With constant multiple

∫(2eˣ - 3cos(x))dx = 2eˣ - 3sin(x) + C

Practice Problems

Problem 1: ∫x⁴dx

Problem 2: ∫(3x² + 2x)dx

Problem 3: ∫e^(2x)dx

Problem 4: ∫cos(3x)dx

Problem 5: ∫(1/x)dx

Problem 6: ∫(5x³ - 4x + 2)dx

Summary

Key takeaways:

• Antiderivative: F'(x) = f(x)
• ∫f(x)dx = F(x) + C (all antiderivatives)
• Power rule: ∫xⁿdx = x^(n+1)/(n+1) + C
• Special: ∫(1/x)dx = ln|x| + C
• ∫eˣdx = eˣ + C
• ∫sin(x)dx = -cos(x) + C, ∫cos(x)dx = sin(x) + C
• Always include + C for indefinite integrals

Pitfalls

• Forgetting the +C constant of integration. An indefinite integral represents a family of functions. Leaving off +C is technically wrong and will cost you points on exams — it's the most common integration mistake.
• Applying the power rule incorrectly. ∫xⁿdx = x^(n+1)/(n+1) + C, NOT x^(n-1)/n + C. The exponent increases by 1 then divides, the opposite of differentiation where the exponent decreases by 1 then multiplies.
• Treating ∫(1/x)dx as x⁰/0. The power rule has a special exception at n = -1: ∫x⁻¹dx = ln|x| + C, not x⁰/0. This catches many students.
• Mixing up signs on trig integrals. ∫sin(x)dx = -cos(x) + C (negative!), while ∫cos(x)dx = sin(x) + C (positive). It's easy to swap these signs because differentiation does the reverse: cos' = -sin, sin' = cos.
• Forgetting to account for the inner derivative. ∫e^(2x)dx = (1/2)e^(2x) + C, not just e^(2x) + C. When the argument is not just x, integration by substitution is needed to handle the chain factor.

Quiz

Q1: ∫x³dx equals:

A) 3x² + C B) x⁴/4 + C C) x² + C D) 4x³ + C

Q2: ∫(2x + 3)dx equals:

A) x² + 3x + C B) 2x² + 3x + C C) x² + 3 + C D) 2x + 3 + C

Q3: ∫eˣdx equals:

A) eˣ + C B) xeˣ + C C) eˣ/x + C D) ln(x) + C

Q4: ∫(1/x)dx equals:

A) x²/2 + C B) ln|x| + C C) 1/x² + C D) eˣ + C

Q5: ∫sin(x)dx equals:

A) cos(x) + C B) -cos(x) + C C) -sin(x) + C D) tan(x) + C

Next Steps

Next up: 05-02-the-definite-integral.md