- Define and explain Introducing Integration in your own words
- Use key terms such as Function accurately
- Apply what you have learned to new examples and questions
- Avoid the common mistakes learners make with this topic
This lesson focuses on Introducing Integration: antidifferentiation, definite integrals, and areas under curves.
Antidifferentiation, definite integrals, and areas under curves.
Key ideas
Integration as reverse differentiation
The integral of xⁿ is xⁿ⁺¹/(n+1) plus a constant C. A definite integral between limits gives the signed area under the curve: ∫₀² x² dx = [x³/3]₀² = 8/3.
Differentiating polynomials
From first principles, the gradient of xⁿ is nxⁿ⁻¹ — the power rule. So f(x) = 3x³ − 5x² + 2x − 7 gives f′(x) = 9x² − 10x + 2, and at x = 2 the gradient is 36 − 20 + 2 = 18.
Key term — Function: A rule that assigns each input exactly one output, written y = f(x); the set of allowed inputs is the domain.
Given f(x) = 3x³ − 5x² + 2x − 7, find f′(x) and the gradient at x = 2.
Apply the power rule to each term: 3x³ → 9x², −5x² → −10x, 2x → 2, and the constant −7 differentiates to 0. So f′(x) = 9x² − 10x + 2. Substitute x = 2: 9(4) − 10(2) + 2 = 36 − 20 + 2 = 18.
Answer: f′(x) = 9x² − 10x + 2; gradient at x = 2 is 18.
- Losing the constant of integration An indefinite integral must end with + C; different values of C give the whole family of antiderivatives.
- Differentiating the constant term to 1 or x The derivative of any constant is 0 — it contributes no change.
Practice
dy/dx = 12x² + 6.
5 + 9 × 4 = 5 + 36 = 41.
2x³ + 4x + C.
[x²]₀³ = 9 − 0 = 9.
Quick check
Which of these best defines "Function"?
Find f(g(x)) given f(x) = x² and g(x) = 2x + 1.
Find the stationary points of y = x² − 6x + 5.
- Introducing Integration: antidifferentiation, definite integrals, and areas under curves.
- Integration as reverse differentiation: The integral of xⁿ is xⁿ⁺¹/(n+1) plus a constant C.
- Arithmetic progression: A sequence where each term differs from the previous by a constant common difference, e.g.
- Watch out for: losing the constant of integration