CHAPTER 03 — INTEGRATION | PART 1: FOUNDATIONS AND ELEMENTARY INTEGRATION

MATHEMATICS HIP-HOP ACADEMY • A-LEVEL PURE MATHEMATICS

CHAPTER 03 — INTEGRATION

PART 1: FOUNDATIONS AND ELEMENTARY INTEGRATION

Integration is a central idea of calculus. It reverses differentiation and provides methods for finding accumulated quantities such as areas and total changes.

1. Learning Objectives

By the end of this lesson, you should be able to:

• Explain integration as the reverse of differentiation.
• Distinguish indefinite and definite integrals.
• Apply the basic rules of integration.
• Integrate algebraic expressions involving positive, fractional and negative powers.
• Find the constant of integration using a given condition.
• Evaluate simple definite integrals.

2. Meaning of Integration

Differentiation finds the rate at which a function changes. Integration reverses this process: it recovers a function from its derivative.

If dy/dx = f(x), then y = ∫ f(x) dx

For example, since the derivative of x³ is 3x², an integral of 3x² is x³.

Remember: Differentiation moves from a function to its derivative; integration works backwards.

3. Indefinite Integrals and the Constant C

An indefinite integral gives a family of functions. We include + C because the derivative of a constant is zero.

∫ f(x) dx = F(x) + C

For example, x² + 1, x² − 7 and x² + 100 all have derivative 2x. Therefore:

∫ 2x dx = x² + C

Always include the constant of integration when evaluating an indefinite integral.

4. Basic Rules of Integration

A. Power Rule

∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, where n ≠ −1

Add one to the power, then divide by the new power.

B. Constant Rule

∫ k dx = kx + C

C. Constant Multiple Rule

∫ kf(x) dx = k∫f(x) dx

D. Sum and Difference Rules

∫[f(x) ± g(x)]dx = ∫f(x)dx ± ∫g(x)dx

Integrate each term separately.

5. Worked Examples

Example 1: A Simple Power

Evaluate ∫x⁴ dx.

Increase the power: 4 + 1 = 5. Divide by 5.
∫x⁴ dx = x⁵/5 + C.

Answer: x⁵/5 + C

Example 2: A Polynomial

Evaluate ∫(6x² − 4x + 7)dx.

∫6x²dx = 2x³
∫−4x dx = −2x²
∫7dx = 7x
Combine the terms and add C.

Answer: 2x³ − 2x² + 7x + C

Example 3: Fractional Powers

Evaluate ∫(3√x + 2/√x)dx.

Rewrite using indices: 3x^1/2 + 2x^−1/2.
Integrate: 3 × x^3/2/(3/2) + 2 × x^1/2/(1/2) + C.

Answer: 2x^3/2 + 4x^1/2 + C

Example 4: Negative Powers

Evaluate ∫(5x⁻³ − 4x⁻²)dx.

5 × x⁻²/(−2) − 4 × x⁻¹/(−1) + C

Answer: −5/(2x²) + 4/x + C

Example 5: Finding C

Given dy/dx = 6x − 4 and y = 5 when x = 2, find y.

Integrate: y = 3x² − 4x + C.
Substitute x = 2 and y = 5:
5 = 12 − 8 + C, so C = 1.

Answer: y = 3x² − 4x + 1

6. Definite Integration

A definite integral has a lower and an upper limit. It gives a numerical value.

∫ₐᵇ f(x)dx = [F(x)]ₐᵇ = F(b) − F(a)

Find an antiderivative, substitute the upper limit, then subtract the value at the lower limit.

Example 6

Evaluate ∫₁³ 2x dx.

Antiderivative: x².
[x²]₁³ = 3² − 1² = 9 − 1 = 8.

Answer: 8

Example 7

Evaluate ∫₀²(3x² + 2)dx.

Antiderivative: x³ + 2x.
[x³ + 2x]₀² = (8 + 4) − 0 = 12.

Answer: 12

A definite integral gives signed area. Portions below the x-axis contribute negative values. Total geometric area may require splitting the interval.

7. The Special Integral of 1/x

The power rule cannot be used for x⁻¹. Instead:

∫(1/x)dx = ln|x| + C, where x ≠ 0

Example 8

Evaluate ∫(4/x − 3x²)dx.

4∫(1/x)dx − 3∫x²dx = 4ln|x| − x³ + C.

Answer: 4ln|x| − x³ + C

8. Common Mistakes

• Forgetting to increase the power before dividing.
• Dividing by the original power instead of the new power.
• Leaving out + C in an indefinite integral.
• Using the power rule for 1/x.
• Forgetting to use a given condition to calculate C.
• Reversing the order when applying definite limits.
• Assuming a definite integral always equals total geometric area.

9. Practice Exercises

Attempt each question before revealing its answer.

1. Evaluate ∫7x⁶ dx.

Reveal answer

x⁷ + C

2. Evaluate ∫(8x³ − 6x + 5)dx.

Reveal answer

2x⁴ − 3x² + 5x + C

3. Evaluate ∫(4√x + 3/x²)dx.

Reveal answer

(8/3)x^3/2 − 3/x + C

4. Evaluate ∫(6x⁻⁴ + 2x^−1/2)dx.

Reveal answer

−2x⁻³ + 4x^1/2 + C

5. Find y if dy/dx = 10x − 6 and y = 3 when x = 1.

Reveal answer

y = 5x² − 6x + 4

6. Evaluate ∫₀² 3x² dx.

Reveal answer

8

7. Evaluate ∫₁³(2x + 1)dx.

Reveal answer

10

8. Evaluate ∫(5/x + 4x²)dx.

Reveal answer

5ln|x| + (4/3)x³ + C

9. Find ∫(3x² − 4x + 2)dx and verify by differentiation.

Reveal answer

x³ − 2x² + 2x + C. Differentiating gives 3x² − 4x + 2.

10. Evaluate ∫₀¹(4x³ + 2x)dx.

Reveal answer

2

10. Lesson Summary

• Integration reverses differentiation.
• The power rule increases the power by one and divides by the new power, for n ≠ −1.
• Indefinite integrals include + C.
• ∫(1/x)dx = ln|x| + C.
• Definite integrals are evaluated using F(b) − F(a).

NEXT: PART 2 — SUBSTITUTION AND RECOGNISING A FUNCTION WITH ITS DERIVATIVE

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