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📑 In This Chapter Guide (Table of Contents)
1. Standard Differentiation Formulas & Rules
| Function f(x) | Derivative d/dx [f(x)] | Inverse Trig Functions | Derivative |
|---|---|---|---|
xⁿ |
n · xⁿ⁻¹ |
sin⁻¹(x) |
1 / √(1 - x²) |
sin(x) |
cos(x) |
cos⁻¹(x) |
-1 / √(1 - x²) |
cos(x) |
-sin(x) |
tan⁻¹(x) |
1 / (1 + x²) |
tan(x) |
sec²(x) |
cot⁻¹(x) |
-1 / (1 + x²) |
eˣ |
eˣ |
sec⁻¹(x) |
1 / [ |x| · √(x² - 1) ] |
ln(x) |
1 / x |
aˣ |
aˣ · ln(a) |
Product Rule: d/dx (u · v) = u · (dv/dx) + v · (du/dx)
Quotient Rule: d/dx (u / v) = [ v · (du/dx) - u · (dv/dx) ] / v²
2. Standard Indefinite Integrals & Special Integrals
∫ xⁿ dx = [ x^(n+1) / (n+1) ] + C(n ≠ -1)∫ (1/x) dx = ln|x| + C;∫ eˣ dx = eˣ + C∫ sin(x) dx = -cos(x) + C;∫ cos(x) dx = sin(x) + C∫ sec²(x) dx = tan(x) + C;∫ cosec²(x) dx = -cot(x) + C∫ tan(x) dx = ln|sec x| + C = -ln|cos x| + C∫ cot(x) dx = ln|sin x| + C
Special Integrals (Crucial for 3-mark & 5-mark integrals):
∫ dx / (x² + a²) = (1/a) · tan⁻¹(x/a) + C∫ dx / (x² - a²) = [ 1 / (2a) ] · ln | (x - a) / (x + a) | + C∫ dx / (a² - x²) = [ 1 / (2a) ] · ln | (a + x) / (a - x) | + C∫ dx / √(a² - x²) = sin⁻¹(x/a) + C∫ dx / √(x² ± a²) = ln | x + √(x² ± a²) | + C
3. Integration by Parts & Definite Integral Properties
Integration by Parts (ILATE Priority Rule: Inverse, Logarithmic, Algebraic, Trigonometric, Exponential):
∫ u · v dx = u · ∫ v dx - ∫ [ (du/dx) · ∫ v dx ] dx
Classic eˣ Shortcut: ∫ eˣ [ f(x) + f'(x) ] dx = eˣ · f(x) + C
King’s Property of Definite Integrals (Used in 90% of exam proofs):
∫₀ᵃ f(x) dx = ∫₀ᵃ f(a - x) dx
∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b - x) dx
💡 Frequently Asked Questions (FAQ)
❓ Evaluate ∫₀^(π/2) [ √(sin x) / (√(sin x) + √(cos x)) ] dx using King’s property.
Let I = ∫₀^(π/2) [ √(sin x) / (√(sin x) + √(cos x)) ] dx ... (1). By King’s Property ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a-x)dx, replace x by (π/2 - x): I = ∫₀^(π/2) [ √(cos x) / (√(cos x) + √(sin x)) ] dx ... (2). Adding (1) and (2): 2I = ∫₀^(π/2) 1 dx = [x]₀^(π/2) = π/2. Therefore: I = π/4.
❓ What is the integrating factor (I.F.) of linear differential equation dy/dx + P(x)·y = Q(x)?
The integrating factor is I.F. = e^(∫ P(x) dx). The general solution of the differential equation is then given by: y · (I.F.) = ∫ [ Q(x) · (I.F.) ] dx + C.
📚 Related Study Guides & Notes
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