Integration

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Integration is the inverse process of differentiation. If F'(x) = f(x), then ∫f(x)dx = F(x) + C, where C is the constant of integration. Example: Since d/dx(x²) = 2x, we have ∫2x dx = x² + C Key Poi

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The constant of integration 'C' is added because: 1. Differentiation of any constant is zero 2. If F(x) is an antiderivative of f(x), then F(x) + C is also an antiderivative Example: • d/dx(x² + 5)

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∫x^n dx = x^(n+1)/(n+1) + C, where n ≠ -1 Step-by-step verification: • Let F(x) = x^(n+1)/(n+1) • Then F'(x) = (n+1)x^n/(n+1) = x^n ✓ Examples: 1. ∫x³ dx = x⁴/4 + C 2. ∫x^(1/2) dx = x^(3/2)/(3/2) =

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∫(1/x) dx = log|x| + C Why it's special: • The power rule ∫x^n dx = x^(n+1)/(n+1) + C fails when n = -1 • This gives us 0/0 form, which is undefined • Instead, we use the fact that d/dx(log|x|) = 1/x

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Standard Trigonometric Integrals: 1. ∫sin x dx = -cos x + C 2. ∫cos x dx = sin x + C 3. ∫sec² x dx = tan x + C 4. ∫cosec² x dx = -cot x + C 5. ∫sec x tan x dx = sec x + C 6. ∫cosec x cot x dx = -cose

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Exponential Functions: 1. ∫e^x dx = e^x + C 2. ∫a^x dx = a^x/log a + C (a > 0, a ≠ 1) Inverse Trigonometric Functions: 3. ∫1/√(1-x²) dx = sin⁻¹x + C 4. ∫1/(1+x²) dx = tan⁻¹x + C 5. ∫1/(x√(x²-1)) dx =

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Properties of Indefinite Integrals: 1. ∫[f(x) ± g(x)]dx = ∫f(x)dx ± ∫g(x)dx (Linearity Property) 2. ∫k·f(x)dx = k∫f(x)dx (where k is a constant) (Constant Multiple Property) Example Applicati

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Step-by-step Solution using Substitution: Step 1: Let u = 2x + 3 Step 2: Differentiate: du = 2dx, so dx = du/2 Step 3: Substitute: ∫(2x + 3)⁵ dx = ∫u⁵ · (du/2) = (1/2)∫u⁵ du Step 4: Integrate: = (1/