Definite Integration: One-Page Formula Sheet

Definite Integration — formula sheet

Formulas

• ∫₋ₐᵃ f = 2∫₀ᵃ f if f even, 0 if odd
• ∫₀^(nT) f = n∫₀ᵀ f for period T
• Leibniz: d/dx ∫_{g(x)}^{h(x)} f(t) dt = f(h)h′ − f(g)g′
• Limit of a sum: lim (1/n)Σ f(r/n) = ∫₀¹ f(x) dx

Key ideas

• ∫ₐᵇ f(x) dx = F(b) − F(a)
• King's property: ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b − x) dx
• Odd function over [−a, a] integrates to 0
• Leibniz rule for derivative of an integral

Traps to avoid

• Not splitting at points where |f| or [x] changes
• Forgetting to change limits after substitution
• Sign errors in Leibniz rule

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