Types of Integrals — Definite, Indefinite & Improper

Indefinite Integrals

∫f(x)dx = F(x)+C. Asks: what function has derivative f? Answer is a family of functions. Used in solving differential equations, finding general antiderivatives. Example: ∫3x²dx = x³+C.

Definite Integrals

∫ₐᵇ f(x)dx = F(b)−F(a). A specific number — the net signed area under f from a to b. 'Signed' means area below the x-axis subtracts. Used in area, volume, distance, and accumulation problems.

Improper Integrals

• An improper integral has an infinite bound or an unbounded integrand:
• Type 1: ∫₁^∞ 1/x² dx — upper bound is ∞. Replace ∞ with t, integrate, take limit as t→∞.
• Type 2: ∫₀¹ 1/√x dx — integrand blows up at x=0. Replace 0 with ε, integrate, take limit as ε→0.
• Result: converges if limit is finite; diverges if not.

Computing Improper Integrals

• ∫₁^∞ 1/x² dx = lim(t→∞) ∫₁ᵗ x⁻² dx = lim(t→∞) [−x⁻¹]₁ᵗ = lim(t→∞) (−1/t + 1) = 0+1 = 1.
• ∫₁^∞ 1/x dx = lim(t→∞) [ln x]₁ᵗ = lim ln(t) → ∞ (diverges).

The Conceptual Distinction — Why Three Types?

The three types of integrals are not just notational variants — they answer fundamentally different questions. The indefinite integral asks: what function has derivative f? The answer is a family of functions, not a number. The definite integral asks: what is the total accumulation of f from a to b? The answer is a specific number. The improper integral extends the definite integral to situations where the standard Riemann sum definition fails — unbounded regions or unbounded functions — by introducing limits.

Indefinite Integrals — The +C Is Not Optional

Every indefinite integral includes a constant of integration +C. This is not a formality — it reflects a genuine mathematical fact. The derivative of x² + 5 is 2x. So is the derivative of x² + 100, or x² − 7, or x² + π. Every function of the form x² + C has derivative 2x. Therefore ∫2x dx = x² + C, where C represents any real constant. If you drop the +C, your antiderivative is technically a specific function, not the general antiderivative. On AP exams and university assessments, omitting +C on indefinite integrals loses marks.

The +C becomes crucial when solving initial value problems. If you are told that F'(x) = 2x and F(3) = 14, then F(x) = x² + C and 14 = 9 + C gives C = 5, so F(x) = x² + 5. Without +C in the general solution, you cannot incorporate the initial condition.

Definite Integrals — Interpretation and Computation

The definite integral ∫ₐᵇ f(x)dx has four equivalent interpretations: (1) the limit of Riemann sums, (2) the net signed area under f from a to b, (3) F(b) − F(a) for any antiderivative F (FTC Part 2), and (4) the total accumulation of f over the interval. In applications, which interpretation to use depends on context. Computing it always uses interpretation (3) — find an antiderivative and subtract.

Properties of Definite Integrals

• Reversal: ∫ₐᵇ f dx = −∫ᵦₐ f dx. Swapping bounds negates the integral.
• Zero width: ∫ₐᵃ f dx = 0. Integral over a single point is always zero.
• Additivity: ∫ₐᶜ f dx = ∫ₐᵇ f dx + ∫ᵦᶜ f dx for any b between a and c.
• Linearity: ∫[αf + βg]dx = α∫f dx + β∫g dx.
• Comparison: If f(x) ≥ g(x) on [a,b], then ∫ₐᵇ f dx ≥ ∫ₐᵇ g dx.

Improper Integrals — Intuition and Technique

The standard definition of the Riemann integral requires a closed, bounded interval [a,b] and a bounded function. Improper integrals extend integration beyond these constraints using limits. There are two types: infinite bounds (∫₁^∞) and unbounded integrands (∫₀¹ 1/√x dx, where the integrand approaches infinity at x = 0).

The key insight: just because a function grows without bound does not mean the area under it is infinite. The function 1/x² blows up at x = 0, but ∫₁^∞ 1/x² dx = 1. The tail of the function shrinks fast enough that the total area remains finite. Contrast this with 1/x, which also grows without bound but whose integral diverges: ∫₁^∞ 1/x dx = ∞. The difference between convergence and divergence can be subtle — and the p-integral test (convergence of ∫₁^∞ 1/xᵖ dx depends on whether p > 1) is the single most useful tool for making the determination quickly.

The Fundamental Theorem — Connecting All Three Types

The FTC is what makes all three types of integrals part of the same theory. The indefinite integral gives antiderivatives. The definite integral uses antiderivatives to compute areas via FTC Part 2. Improper integrals extend definite integrals using limits — and when they converge, the FTC applies to them as well, through the limiting process. The entire edifice of integral calculus rests on the FTC.