Integral Rules
Table of Contents
1. What is Integration
Integration is one of the two fundamental operations of calculus, serving as the inverse of differentiation. It comes in two forms: the indefinite integral (finding antiderivatives) and the definite integral (computing the area under a curve).
Indefinite Integral -- the Antiderivative
The indefinite integral answers the question: "Which function has f(x) as its derivative?" If F'(x) = f(x), then F(x) is an antiderivative of f(x). Since the derivative of any constant is zero, antiderivatives always include an arbitrary constant C:
Definition of the Indefinite Integral
∫ f(x) dx = F(x) + C, where F'(x) = f(x)
Definite Integral -- Area Under the Curve
The definite integral computes the "signed area" of the region between f(x) and the x-axis over the interval [a, b]. Riemann (Bernhard Riemann, 1854) rigorously defined it as the limit of sums of rectangle areas:
Definition (Riemann Sum)
∫ₚᵇ f(x) dx = lim(n→∞) Σᵢ₀ⁿ f(xᵢ*) · Δx
where Δx = (b-a)/n and xᵢ* is a sample point in the i-th subinterval.
2. Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus (FTC) is the most important theorem in all of calculus. It reveals that differentiation and integration -- two seemingly unrelated operations -- are actually inverse operations of each other.
Part 1 (Differentiation Undoes Integration)
d/dx [ ∫ₚˣ f(t) dt ] = f(x)
Meaning: if you integrate f and then differentiate, you get back to f. The derivative of the integral with a variable upper limit equals the integrand.
Part 2 (The Newton-Leibniz Formula)
∫ₚᵇ f(x) dx = F(b) - F(a), where F'(x) = f(x)
Meaning: a definite integral can be evaluated by finding the antiderivative and computing the difference at the endpoints -- no need to take limits of Riemann sums!
History: Newton and Leibniz independently discovered this relationship in the latter half of the 17th century. Newton's teacher Isaac Barrow (1630-1677) already understood the geometric inverse relationship between differentiation and integration, but did not provide an analytic proof. Augustin-Louis Cauchy (1823) gave the first rigorous analytical proof.
3. Basic Integration Rules
Each rule below is derived by "reversing" a known derivative rule -- every integration rule corresponds to a differentiation rule.
| Rule | Formula | Why It Works |
|---|---|---|
| Constant Rule | ∫ c dx = cx + C |
Because d/dx[cx] = c |
| Power Rule | ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1) |
Reverse of the power rule for derivatives: d/dx[xⁿ⁺¹/(n+1)] = xⁿ. This is the most commonly used integration rule. |
| Reciprocal Rule | ∫ 1/x dx = ln|x| + C |
Fills the gap where the power rule fails (n = -1). Because d/dx[ln|x|] = 1/x. |
| Exponential Rule | ∫ eˣ dx = eˣ + C |
Because e is defined so that eˣ is its own derivative. Euler's number e ≈ 2.71828 is the unique base with this property. |
| General Exponential | ∫ aˣ dx = aˣ/ln(a) + C (a > 0, a ≠ 1) |
Because d/dx[aˣ] = aˣ · ln(a); dividing by ln(a) gives the antiderivative. |
| Sum/Difference Rule | ∫ [f(x) ± g(x)] dx = ∫f dx ± ∫g dx |
Reverse of the linearity of derivatives: (F ± G)' = F' ± G'. |
| Constant Multiple | ∫ c · f(x) dx = c · ∫f(x) dx |
Constants can be pulled out of the integral because d/dx[c·F] = c·F'. |
Logarithmic Integral
∫ ln(x) dx = x·ln(x) - x + C
Derived via integration by parts: let u = ln(x), dv = dx.
4. Trigonometric Integrals Table
All trigonometric integral formulas can be verified by differentiating the result.
| Integrand f(x) | ∫ f(x) dx | Verification / Notes |
|---|---|---|
| sin(x) | -cos(x) + C | d/dx[-cos(x)] = sin(x) |
| cos(x) | sin(x) + C | d/dx[sin(x)] = cos(x) |
| tan(x) | -ln|cos(x)| + C = ln|sec(x)| + C | Let u = cos(x), reduces to -∫du/u |
| cot(x) | ln|sin(x)| + C | Let u = sin(x) |
| sec(x) | ln|sec(x) + tan(x)| + C | Classic trick: multiply by (sec(x)+tan(x))/(sec(x)+tan(x)) |
| csc(x) | -ln|csc(x) + cot(x)| + C | Analogous to sec(x) method |
| sec²(x) | tan(x) + C | d/dx[tan(x)] = sec²(x) |
| csc²(x) | -cot(x) + C | d/dx[-cot(x)] = csc²(x) |
| sec(x)·tan(x) | sec(x) + C | d/dx[sec(x)] = sec(x)·tan(x) |
| csc(x)·cot(x) | -csc(x) + C | d/dx[-csc(x)] = csc(x)·cot(x) |
Inverse Trigonometric Integrals
| Integrand | Result |
|---|---|
| 1/√(1 - x²) | arcsin(x) + C |
| -1/√(1 - x²) | arccos(x) + C |
| 1/(1 + x²) | arctan(x) + C |
| 1/(x√(x² - 1)) | arcsec(|x|) + C |
5. Integration Techniques
Not all functions can be integrated using the basic rules directly. The following four advanced techniques are the core tools for tackling complex integrals.
5.1 U-Substitution
Formula
∫ f(g(x)) · g'(x) dx = ∫ f(u) du, where u = g(x)
Steps: 1) Identify the inner function g(x) and set u = g(x). 2) Compute du = g'(x) dx. 3) Rewrite the entire integral in terms of u. 4) Integrate, then substitute back u = g(x).
5.2 Integration by Parts
Formula
∫ u dv = uv - ∫ v du
The LIATE Rule for choosing u:
L -- Logarithmic: ln(x), log(x)
I -- Inverse trigonometric: arcsin(x), arctan(x)
A -- Algebraic: x², x, √x
T -- Trigonometric: sin(x), cos(x)
E -- Exponential: eˣ, 2ˣ
Why this order works: Functions higher on the list become simpler when differentiated (e.g., ln(x) → 1/x), while functions lower on the list remain manageable when integrated (e.g., eˣ → eˣ).
5.3 Partial Fraction Decomposition
Idea
Decompose a rational function P(x)/Q(x) into a sum of simpler fractions, each of which has a known antiderivative.
Example: ∫ (2x+3)/((x+1)(x+2)) dx = ∫ [1/(x+1) + 1/(x+2)] dx = ln|x+1| + ln|x+2| + C
When to use: The integrand is a ratio of two polynomials (a rational function) and the denominator can be factored.
5.4 Trigonometric Substitution
| Integrand Contains | Substitute | Identity Used |
|---|---|---|
| √(a² - x²) | x = a·sin(θ) | 1 - sin²θ = cos²θ → √(a² - x²) = a·cos(θ) |
| √(a² + x²) | x = a·tan(θ) | 1 + tan²θ = sec²θ → √(a² + x²) = a·sec(θ) |
| √(x² - a²) | x = a·sec(θ) | sec²θ - 1 = tan²θ → √(x² - a²) = a·tan(θ) |
6. Definite Integral Properties
The definite integral possesses several important algebraic properties that are invaluable for simplifying computations and constructing proofs.
| Property | Formula |
|---|---|
| Linearity | ∫ₚᵇ [αf(x) + βg(x)] dx = α∫ₚᵇ f dx + β∫ₚᵇ g dx |
| Additivity | ∫ₚᵇ f dx + ∫ᵇᶜ f dx = ∫ₚᶜ f dx |
| Reversal | ∫ₚᵇ f dx = -∫ᵇₚ f dx |
| Zero Width | ∫ₚᵃ f dx = 0 |
| Comparison | If f(x) ≥ g(x) on [a,b], then ∫ₚᵇ f dx ≥ ∫ₚᵇ g dx |
| Triangle Inequality | |∫ₚᵇ f dx| ≤ ∫ₚᵇ |f(x)| dx |
Mean Value Theorem for Integrals
If f is continuous on [a,b], there exists c ∈ [a,b] such that ∫ₚᵇ f(x) dx = f(c) · (b - a)
Intuition: a continuous function's "average value" over an interval is actually attained at some point.
7. Improper Integrals
When the interval of integration is unbounded (extends to ±∞) or the integrand is unbounded at some point (has a vertical asymptote), the resulting integral is called an improper integral.
Type I: Infinite Limits
∫ₚ^∞ f(x) dx = lim(b→∞) ∫ₚᵇ f(x) dx
If the limit exists and is finite, the integral converges; otherwise it diverges.
Type II: Unbounded Integrand
If f is unbounded at x = a: ∫ₚᵇ f(x) dx = lim(ε→0⁺) ∫ₚ₊εᵇ f(x) dx
Convergence Tests
Comparison Test: If 0 ≤ f(x) ≤ g(x) and ∫g converges, then ∫f also converges.
Limit Comparison Test: If lim(x→∞) f(x)/g(x) = L (0 < L < ∞), then ∫f and ∫g both converge or both diverge.
8. Applications of Integration
8.1 Area Between Curves
A = ∫ₚᵇ |f(x) - g(x)| dx
Simplifies to ∫ₚᵇ [f(x) - g(x)] dx when f(x) ≥ g(x).
8.2 Volumes of Revolution
Disk Method
V = π ∫ₚᵇ [f(x)]² dx
Rotation about the x-axis; cross-sections are disks.
Shell Method
V = 2π ∫ₚᵇ x · f(x) dx
Rotation about the y-axis; uses concentric cylindrical shells.
8.3 Arc Length
L = ∫ₚᵇ √(1 + [f'(x)]²) dx
Length of the curve y = f(x) from x = a to x = b.
8.4 Probability and Integration
Normalization: ∫∞∞ f(x) dx = 1
Interval probability: P(a ≤ X ≤ b) = ∫ₚᵇ f(x) dx
Expected value: E[X] = ∫∞∞ x · f(x) dx
8.5 Integration in Physics
Center of mass: x̄ = (1/M) ∫ₚᵇ x · ρ(x) dx, where M = ∫ₚᵇ ρ(x) dx
Fluid pressure: F = ∫ₚᵇ ρg · h(y) · w(y) dy
9. Famous Integrals
The Gaussian Integral
∫∞∞ e^(-x²) dx = √π
Equivalent form: ∫₀^∞ e^(-x²) dx = √π/2
The elegant proof: Let I = ∫∞∞ e^(-x²) dx. Then I² = ∫∫ e^(-(x²+y²)) dx dy. Convert to polar coordinates (r, θ) using x² + y² = r²: the integral becomes ∫₀^(2π) ∫₀^∞ e^(-r²) · r dr dθ = 2π · (1/2) = π, so I = √π.
The Dirichlet Integral
∫₀^∞ sin(x)/x dx = π/2
History: Named after Peter Gustav Lejeune Dirichlet (1805-1859), who used it in his rigorous study of the convergence of Fourier series.
10. Related Tools
- Derivative Rules Reference -- differentiation is the inverse of integration
- Limits Reference -- the rigorous definition of integrals is based on limits
- Probability Calculator -- probability density integration
11. FAQ
The indefinite integral ∫f(x)dx finds the antiderivative -- a family of functions (with constant C). The definite integral ∫ₚᵇf(x)dx computes the "signed area" under the curve -- a specific number. The Fundamental Theorem of Calculus connects them: the definite integral equals the antiderivative evaluated at the endpoints.
Because the derivative of any constant is zero. If F(x) is one antiderivative of f(x), then F(x) + 5, F(x) - 3, etc. are all also antiderivatives. The constant C represents any member of this family. In definite integrals, C cancels out: F(b)+C - (F(a)+C) = F(b) - F(a).
LIATE is a heuristic that works in the vast majority of cases, but it is not absolute. For example, in ∫xⁿ·ln(x)dx, choosing u = ln(x) per LIATE does simplify the problem. However, some special cases may require flexibility. The key principle remains: choose u so that du is simpler, and dv so that v is easy to find.
Yes -- by Part 1 of the FTC, every continuous function f(x) has an antiderivative F(x) = ∫ₚˣ f(t)dt. However, this does not mean the antiderivative can always be expressed using elementary functions. For example, the antiderivative of e^(-x²) (the error function erf) cannot be written as a finite combination of elementary functions. Joseph Liouville (1809-1882) developed the theory for determining whether an integral can be expressed in elementary terms.
An improper integral converges when its value approaches a finite number as the limit of integration goes to infinity (or approaches the singular point). For example, ∫₁^∞ 1/x² dx = 1 (converges), but ∫₁^∞ 1/x dx = ∞ (diverges). Intuitively, the integrand must decay fast enough for the integral to converge. The p-test gives the precise threshold: 1/xᵖ converges for p > 1.