Definite Integration

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A definite integral is the difference φ(b) - φ(a) where φ'(x) = f(x), denoted as ∫ᵃᵇ f(x) dx = φ(b) - φ(a). Unlike indefinite integrals, definite integrals: • Have specific limits of integration (a an

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If f is continuous on [a,b] and ∫f(x)dx = φ(x) + c, then: ∫ᵃᵇ f(x) dx = [φ(x)]ᵃᵇ = φ(b) - φ(a) Example: Evaluate ∫₂³ x⁴ dx Solution: • φ(x) = x⁵/5 • ∫₂³ x⁴ dx = [x⁵/5]₂³ = 3⁵/5 - 2⁵/5 = 243/5 - 32/5

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Step-by-step solution: ∫₀¹ 1/(2x+5) dx Step 1: Recognize this as ∫ 1/(ax+b) dx = (1/a)ln|ax+b| + c Step 2: Here a = 2, b = 5 Step 3: ∫₀¹ 1/(2x+5) dx = (1/2)[ln|2x+5|]₀¹ Step 4: = (1/2)[ln|2(1)+5| - l

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∫ₐᵃ f(x) dx = 0 Explanation: When both limits of integration are the same, the integral equals zero because φ(a) - φ(a) = 0. This makes intuitive sense: there is no 'area' between a single point. E

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∫ₐᵇ f(x) dx = -∫ᵇₐ f(x) dx Explanation: Swapping the limits of integration changes the sign. Proof: ∫ₐᵇ f(x) dx = φ(b) - φ(a) ∫ᵇₐ f(x) dx = φ(a) - φ(b) = -(φ(b) - φ(a)) Example: If ∫₁³ x² dx

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∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx, where a < c < b Explanation: An integral over an interval can be split into sum of integrals over subintervals. Use this when: • The function has different d

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This property states that ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b - x) dx Explanation: Substituting x with (a + b - x) doesn't change the integral value. Example: Evaluate ∫₀³ x⁴/(x⁴ + 7⁴) dx Let I = ∫₀³ x⁴/(x⁴ +

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∫₀ᵃ f(x) dx = ∫₀ᵃ f(a - x) dx This is most useful when: • Evaluating integrals with symmetric expressions • The integrand becomes simpler after substitution • Dealing with trigonometric functions Ex