The definite integral and the Fundamental Theorem — A/L Combined Mathematics

What must I remember about The definite integral and the Fundamental Theorem?

• Fundamental Theorem of Calculus: If f(x) is continuous on [a, b] and F'(x) = f(x), then ∫_a^b f(x) dx = F(b) - F(a); connects differential calculus with integral calculus.
• Linearity properties of integrals: ∫ [k1*f(x) ± k2*g(x)] dx = k1 ∫ f(x) dx ± k2 ∫ g(x) dx; ∫_a^a f(x) dx = 0; ∫_a^b f(x) dx = - ∫_b^a f(x) dx.
• Interval additivity: ∫_a^b f(x) dx = ∫_a^c f(x) dx + ∫_c^b f(x) dx for any real number c where f is integrable.
• Definite integral symmetry properties: For even functions (f(-x) = f(x)), ∫_-a^a f(x) dx = 2 ∫₀^a f(x) dx; for odd functions (f(-x) = -f(x)), ∫_-a^a f(x) dx = 0.
• The King's Property: ∫_a^b f(x) dx = ∫_a^b f(a + b - x) dx; in particular, ∫₀^a f(x) dx = ∫₀^a f(a - x) dx; extremely powerful for evaluating symmetric trigonometric integrals.

Test yourself (1 minute)

1. Leibniz Integral Rule for differentiating an integral

   d/dx [∫_u(x)^v(x) f(t) dt] = f(v(x)) * v'(x) - f(u(x)) * u'(x).

2. Integration by Parts for definite integrals

   ∫_a^b u v' dx = [u v]_a^b - ∫_a^b v u' dx; select u using LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential).

3. Periodic function property

   If f(x) has period T (f(x + T) = f(x)), then ∫₀^nT f(x) dx = n ∫₀^T f(x) dx for any integer n.

Practise The definite integral and the Fundamental Theorem questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 15

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Where does The definite integral and the Fundamental Theorem fit in the syllabus?

Year 1 · Term 2