AI Tutor
If someone asks you to find the antiderivative of 7x4−5sin⁡(x)+2ex7x^4 - 5\sin(x) + 2e^x, does that require one monstrous new formula, or can you simply tear it into bite-sized pieces?
Fortunately, integration handles sums and constants just like differentiation does, breaking complex functions into effortless individual steps.
In 1675, Gottfried Wilhelm Leibniz introduced the integral symbol ∫\int as an elongated "S" (for the Latin summa, meaning sum) to show that integration fundamentally accumulates small parts.
An indefinite integral, written ∫f(x) dx\int f(x)\,dx, represents the family of all antiderivatives of f(x)f(x), satisfying ddx[F(x)+C]=f(x)\frac{d}{dx}[F(x) + C] = f(x).
What rules allow us to split complicated expressions apart so easily?

The Two Linearity Properties

A mathematical operation has linearity when it preserves addition and scalar multiplication. For indefinite integrals, this means two core rules.
The constant multiple rule states that for any constant kk, you can factor it out in front of the integral: ∫k f(x) dx=k∫f(x) dx\int k\,f(x)\,dx = k \int f(x)\,dx.
The sum and difference rule states that the integral of a sum or difference equals the sum or difference of the individual integrals: ∫[f(x)±g(x)] dx=∫f(x) dx±∫g(x) dx\int [f(x) \pm g(x)]\,dx = \int f(x)\,dx \pm \int g(x)\,dx.
Why are we allowed to just factor out numbers and split across plus and minus signs?

Why the Rules Work: The Proof

Every integration property is an exact mirror of a derivative property, because finding an indefinite integral is literally undoing a derivative.