H: Integration: Know and use the Fundamental Theorem of Calculus.

The idea

The Fundamental Theorem of Calculus (FTC) links differentiation and integration. It has two parts: (1) if F(x) = ∫_a^x f(t) dt, then F'(x) = f(x); (2) ∫_a^b f(x) dx = F(b) - F(a), where F is any antiderivative of f. This means differentiation and integration are inverse processes. To use it, find an antiderivative of the integrand, then evaluate it at the upper and lower limits and subtract. For example, ∫_0^2 x² dx = [x³/3]_0^2 = 8/3 - 0 = 8/3. The theorem also allows you to differentiate an integral with a variable limit. In exams, you may be asked to evaluate a definite integral, find a derivative of an accumulation function, or prove a result using the FTC. Understanding the theorem is essential for solving problems involving areas under curves and accumulated change.

What earns the marks

• State the theorem correctly: differentiation and integration are inverse operations.
• Use the theorem to evaluate definite integrals by finding an antiderivative and applying limits.
• Differentiate an integral with a variable upper limit correctly.
• Apply the theorem to problems involving area under a curve or accumulated change.
• Recognise when the theorem is applicable and when adjustments (e.g., chain rule) are needed.

Common mistakes and how to put them right

• Forgetting to subtract the value at the lower limit.

  Put it right: Compute F(b) - F(a), not just F(b).

• Incorrect antiderivative: ensure you differentiate back to check. For example, the antiderivative of x² is x³/3, not x³/2.

• Misapplying the theorem when the upper limit is a function of x.

  Put it right: Then you must use the chain rule, e.g., d/dx ∫_a^{g(x)} f(t) dt = f(g(x))g'(x).

Exam tips

• Write the antiderivative in square brackets with limits to show your method clearly.
• Check your answer by differentiating the antiderivative to see if you get the original integrand.
• For variable limits, identify the inner function and apply the chain rule correctly.