Integration Rules
Complete integration formula reference covering the power rule, substitution, integration by parts, and common integrals.
Worked examples for each rule.
Basic Integration Rules
∫ x^n dx = x^(n+1)/(n+1) + C, where n ≠ -1 (power rule)
∫ 1/x dx = ln|x| + C
∫ c·f(x) dx = c · ∫ f(x) dx (constant multiple)
∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx (sum rule)
∫ 1/x dx = ln|x| + C
∫ c·f(x) dx = c · ∫ f(x) dx (constant multiple)
∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx (sum rule)
Common Integrals
| Function | Integral |
|---|---|
| x^n (n ≠ -1) | x^(n+1)/(n+1) + C |
| 1/x | ln|x| + C |
| e^x | e^x + C |
| a^x | a^x / ln(a) + C |
| sin(x) | -cos(x) + C |
| cos(x) | sin(x) + C |
| tan(x) | -ln|cos(x)| + C |
| sec²(x) | tan(x) + C |
| csc²(x) | -cot(x) + C |
| sec(x)·tan(x) | sec(x) + C |
| 1/(1+x²) | arctan(x) + C |
| 1/√(1-x²) | arcsin(x) + C |
| e^(ax) | (1/a)·e^(ax) + C |
Fundamental Theorem of Calculus
∫[a to b] f(x) dx = F(b) - F(a)
Where F(x) is any antiderivative of f(x). This connects differentiation and integration: the definite integral equals the difference of the antiderivative at the bounds.
Integration by Substitution (u-substitution)
∫ f(g(x)) · g'(x) dx = ∫ f(u) du, where u = g(x)
Integration by Parts
∫ u dv = uv - ∫ v du
Choose u and dv using LIATE priority: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential.
Example 1 — Power Rule
Find ∫ 4x³ dx
= 4 · x^(3+1)/(3+1) + C
= 4 · x⁴/4 + C = x⁴ + C
Example 2 — Definite Integral
Find ∫[0 to 2] 3x² dx
Antiderivative: F(x) = x³
= F(2) - F(0) = 8 - 0 = 8
Example 3 — U-Substitution
Find ∫ 2x·cos(x²) dx
Let u = x², then du = 2x dx
= ∫ cos(u) du = sin(u) + C = sin(x²) + C
Key Notes
- Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1): The exception n = −1 gives ∫(1/x)dx = ln|x| + C. Key results: ∫eˣ dx = eˣ + C; ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C; ∫sec²x dx = tan x + C.
- Linearity: ∫(af + bg)dx = a∫f dx + b∫g dx: Constants factor out and sums integrate term by term. This — along with the power rule — handles all polynomial integrals and many combinations of standard functions directly.
- Fundamental Theorem of Calculus: Part 1: d/dx ∫ₐˣ f(t)dt = f(x). Part 2: ∫ₐᵇ f(x)dx = F(b) − F(a) where F is any antiderivative of f. The two operations of differentiation and integration are inverse processes.
- Integration techniques for harder integrals: U-substitution (reverse chain rule); integration by parts: ∫u dv = uv − ∫v du (choose u to simplify on differentiation); partial fractions (rational functions); trigonometric substitution (for √(a²−x²) etc.).
- Applications: Integration computes areas, volumes of revolution, arc lengths, work done by a variable force, probability (area under a PDF), average values, and accumulated change in any quantity described by a rate function (population growth, current flow, heat transfer).