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)
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.
Sum and Difference Rules
Integrals distribute over addition and subtraction:
∫[f(x)+g(x)]dx=∫f(x)dx+∫g(x)dx
∫[f(x)−g(x)]dx=∫f(x)dx−∫g(x)dx
These rules apply to both definite and indefinite integrals. They allow complex integrands to be broken into simpler pieces, each handled separately.
For example:
∫(x3+sinx)dx=∫x3dx+∫sinxdx=4x4−cosx+C
Constant Multiple Rule
Constants factor out of integrals:
∫c⋅f(x)dx=c∫f(x)dx
This holds for any constant c. The rule simplifies computation by separating numerical coefficients from the integration process.
For example:
∫7exdx=7∫exdx=7ex+C
Combined with the sum rule, linearity handles any linear combination:
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] equals the sum of integrals over [a,b], [b,c], and [c,d] for any a<b<c<d.
Reversing Limits of Integration
Swapping the limits of a definite integral negates the result:
∫abf(x)dx=−∫baf(x)dx
This follows from the Riemann sum definition. Integrating from a to b accumulates in one direction; integrating from b to a accumulates in the reverse direction.
A useful consequence: if a=b, the integral vanishes:
∫aaf(x)dx=0
No interval means no accumulation.
Fundamental Theorem of Calculus — Part 1
Define the accumulation function:
F(x)=∫axf(t)dt
Part 1 of the Fundamental Theorem states that if f is continuous, then F is differentiable and:
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 F is any antiderivative of f (meaning F′(x)=f(x)), then:
∫abf(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)ab or [F(x)]ab means F(b)−F(a).
Example:
∫13x2dx=3x313=327−31=326
Why Differentiation and Integration Are Inverses
The two parts of the Fundamental Theorem express inverse relationships.
Part 1: Differentiating an integral recovers the integrand.
dxd∫axf(t)dt=f(x)
Part 2: Integrating a derivative recovers the function (up to boundary values).
∫abF′(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.
Comparison Properties
Inequalities between integrands yield inequalities between integrals.
Nonnegativity: If f(x)≥0 on [a,b], then:
∫abf(x)dx≥0
Monotonicity: If f(x)≤g(x) on [a,b], then:
∫abf(x)dx≤∫abg(x)dx
Bounding: If m≤f(x)≤M on [a,b], then:
m(b−a)≤∫abf(x)dx≤M(b−a)
These properties enable estimation when exact computation is difficult or impossible.