Integration and its Applications

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Integration is the inverse process of differentiation. If d/dx(F(x)) = f(x), then ∫f(x)dx = F(x) + C, where F(x) is called the anti-derivative or primitive of f(x), and C is the constant of integratio…

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Indefinite integral: ∫f(x)dx = F(x) + C, represents a family of curves and includes a constant of integration. Definite integral: ∫ᵇₐf(x)dx = F(b) - F(a), has fixed numerical value and represents the …

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∫xⁿdx = x^(n+1)/(n+1) + C, where n ≠ -1. Special cases: ∫1dx = x + C, ∫(1/x)dx = log|x| + C. This rule is used for integrating polynomial terms.

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∫eˣdx = eˣ + C, ∫aˣdx = aˣ/log a + C, ∫(1/x)dx = log|x| + C. These formulas are essential for integrating exponential and inverse functions commonly found in business applications.

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Use substitution when the integrand contains f(g(x))·g'(x) or when we can simplify by substituting u = g(x). Rule: ∫f(g(x))g'(x)dx = ∫f(u)du where g(x) = u. Common substitutions: √x = t, log x = t, or…

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Using the rule ∫f(ax+b)dx = F(ax+b)/a + C: ∫(2x+3)⁵dx = (2x+3)⁶/(6×2) + C = (2x+3)⁶/12 + C. This uses the linear substitution rule where we divide by the coefficient of x.

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Partial fractions is used to integrate rational functions P(x)/Q(x) where degree of P(x) < degree of Q(x). We decompose the fraction into simpler fractions: (px+q)/((x-a)(x-b)) = A/(x-a) + B/(x-b). Th…

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For repeated factors like (x-a)²: (px+q)/(x-a)² = A/(x-a) + B/(x-a)². For (x-a)³, we need A/(x-a) + B/(x-a)² + C/(x-a)³. Each power of the repeated factor gets its own term in the decomposition.