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Integration Rules



Sum and Difference Rules
Constant Multiple Rule
Additivity Over Intervals
Reversing Limits of Integration
Fundamental Theorem of Calculus — Part 1
Fundamental Theorem of Calculus — Part 2
Why Differentiation and Integration Are Inverses
Comparison Properties
Summary: All Integration Rules at a Glance



The Algebra of Integrals


Integration obeys algebraic rules that mirror those of differentiation. Sums split, constants factor out, and intervals combine. These properties transform complex integrals into manageable pieces.

The deepest result is the Fundamental Theorem of Calculus, which bridges the two faces of integration. Part 1 says differentiation undoes integration. Part 2 says definite integrals can be computed via antiderivatives:

∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\, dx = F(b) - F(a)


This theorem converts area computation—a limiting process involving infinite sums—into simple evaluation of an antiderivative at two points. It explains why mastering indefinite integration unlocks definite integration as well.

Key Terms

Definite Integral— the object these rules govern
Indefinite Integral— linearity applies to both definite and indefinite forms
Antiderivative— the Fundamental Theorem connects antiderivatives to definite integrals

See All Calculus Definitions →


Constant Multiple Rule


Constants factor out of integrals:

∫c⋅f(x) dx=c∫f(x) dx\int c \cdot f(x)\, dx = c \int f(x)\, dx


This holds for any constant cc. The rule simplifies computation by separating numerical coefficients from the integration process.

For example:

∫7ex dx=7∫ex dx=7ex+C\int 7e^x\, dx = 7 \int e^x\, dx = 7e^x + C


Combined with the sum rule, linearity handles any linear combination:

∫[af(x)+bg(x)] dx=a∫f(x) dx+b∫g(x) dx\int [af(x) + bg(x)]\, dx = a\int f(x)\, dx + b\int g(x)\, dx


Additivity Over Intervals


For definite integrals, integration over adjacent intervals combines:

∫abf(x) dx+∫bcf(x) dx=∫acf(x) dx\int_a^b f(x)\, dx + \int_b^c f(x)\, dx = \int_a^c f(x)\, dx


This property allows splitting an integral at any intermediate point. It proves essential for piecewise functions, where different formulas apply on different subintervals.

The property extends naturally: an integral over [a,d][a, d] equals the sum of integrals over [a,b][a, b], [b,c][b, c], and [c,d][c, d] for any a<b<c<da < b < c < d.

Reversing Limits of Integration


Swapping the limits of a definite integral negates the result:

∫abf(x) dx=−∫baf(x) dx\int_a^b f(x)\, dx = -\int_b^a f(x)\, dx


This follows from the Riemann sum definition. Integrating from aa to bb accumulates in one direction; integrating from bb to aa accumulates in the reverse direction.

A useful consequence: if a=ba = b, the integral vanishes:

∫aaf(x) dx=0\int_a^a f(x)\, dx = 0


No interval means no accumulation.

Fundamental Theorem of Calculus — Part 1


Define the accumulation function:

F(x)=∫axf(t) dtF(x) = \int_a^x f(t)\, dt


Part 1 of the Fundamental Theorem states that if ff is continuous, then FF is differentiable and:

F′(x)=f(x)F'(x) = f(x)


Differentiation undoes integration. The rate of change of accumulated area equals the integrand's value at the boundary.

This result establishes that every continuous function has an antiderivative—namely, its accumulation function. The theorem guarantees antiderivatives exist, even when no elementary formula expresses them.

Fundamental Theorem of Calculus — Part 2


If FF is any antiderivative of ff (meaning F′(x)=f(x)F'(x) = f(x)), then:

∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\, dx = F(b) - F(a)


This is the computational engine of integral calculus. Rather than computing limits of Riemann sums, find an antiderivative and evaluate at the endpoints.

The notation F(x)∣abF(x)\Big|_a^b or [F(x)]ab[F(x)]_a^b means F(b)−F(a)F(b) - F(a).

Example:

∫13x2 dx=x33∣13=273−13=263\int_1^3 x^2\, dx = \frac{x^3}{3}\Big|_1^3 = \frac{27}{3} - \frac{1}{3} = \frac{26}{3}


Why Differentiation and Integration Are Inverses


The two parts of the Fundamental Theorem express inverse relationships.

Part 1: Differentiating an integral recovers the integrand.

ddx∫axf(t) dt=f(x)\frac{d}{dx} \int_a^x f(t)\, dt = f(x)


Part 2: Integrating a derivative recovers the function (up to boundary values).

∫abF′(x) dx=F(b)−F(a)\int_a^b F'(x)\, dx = F(b) - F(a)


These are two directions of the same relationship. Integration accumulates; differentiation measures instantaneous rate. Each operation undoes the other, connecting the geometry of area to the algebra of rates of change.
Aspect FTC — Part 1 FTC — Part 2
Statement F(x) = ∫ax f(t) dt  ⇒  F'(x) = f(x) ∫ab f(x) dx = F(b) − F(a)
What it does differentiating an integral recovers the integrand integrating a derivative recovers the function (up to endpoints)
Hypothesis f is continuous on [a, b] F is any antiderivative of f on [a, b]
Main role existence — guarantees every continuous function has an antiderivative computation — turns area into endpoint evaluation
Typical use differentiate accumulation functions; theoretical arguments evaluate definite integrals via F(b) − F(a)

Comparison Properties


Inequalities between integrands yield inequalities between integrals.

Nonnegativity: If f(x)≥0f(x) \geq 0 on [a,b][a, b], then:

∫abf(x) dx≥0\int_a^b f(x)\, dx \geq 0


Monotonicity: If f(x)≤g(x)f(x) \leq g(x) on [a,b][a, b], then:

∫abf(x) dx≤∫abg(x) dx\int_a^b f(x)\, dx \leq \int_a^b g(x)\, dx


Bounding: If m≤f(x)≤Mm \leq f(x) \leq M on [a,b][a, b], then:

m(b−a)≤∫abf(x) dx≤M(b−a)m(b - a) \leq \int_a^b f(x)\, dx \leq M(b - a)


These properties enable estimation when exact computation is difficult or impossible.

Summary: All Integration Rules at a Glance


The rules covered in the sections above fall into three groups — algebraic rules of linearity, properties of the interval of integration, and the Fundamental Theorem in its two parts — capped by the comparison properties that govern inequalities. The master reference below collects every rule on this page in one table, with a third column flagging whether each applies to both definite and indefinite integrals or to definite integrals only. Use it as a single-glance reminder while working problems; each row points back to the section above where the rule is derived and discussed.
Rule Statement Applies to
Sum / difference ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx definite & indefinite
Constant multiple ∫ c · f(x) dx = c · ∫ f(x) dx definite & indefinite
Linearity ∫ [a f(x) + b g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx definite & indefinite
Additivity over intervals ∫ab f dx + ∫bc f dx = ∫ac f dx definite only
Reversing limits ∫ab f dx = − ∫ba f dx definite only
Empty interval ∫aa f dx = 0 definite only
FTC — Part 1 d/dx [ ∫ax f(t) dt ] = f(x), assuming f continuous definite with variable upper limit
FTC — Part 2 ∫ab f(x) dx = F(b) − F(a), for any antiderivative F definite only
Nonnegativity f(x) ≥ 0 on [a, b]  ⇒  ∫ab f dx ≥ 0 definite only
Monotonicity f(x) ≤ g(x) on [a, b]  ⇒  ∫ab f dx ≤ ∫ab g dx definite only
Bounding m ≤ f(x) ≤ M on [a, b]  ⇒  m(b − a) ≤ ∫ab f dx ≤ M(b − a) definite only