freeCodeCamp/guide/english/mathematics/integration-rules/index.md

title
Integration Rules

Integration Rules

As with differentiation, we use a variety of rules when integrating functions. Below are a few of the most common.

The Constant Rule: ∫k dx = kx + C where k is a constant

The Power Rule: ∫ xⁿ dx = + ^xn+1⁄_n+1 + C when n ≠ 1

Exponential Rule: ∫ e^kx dx = ¹⁄_kx e^kx + C where k is a constant

Trigometric Rules:

∫ cos(x) dx = sin(x) + C

∫ sin(x) dx = -cos(x) + C

Sum/Difference Rules:

∫[f(x) + g(x)]dx = ∫f(x)dx + ∫g(x)dx

∫[f(x) - g(x)]dx = ∫f(x)dx - ∫g(x)dx

Physical Calculus Rules:

∫ v dt = x + C, where x is the displacement and v is the velocity

∫ a dt = v + C, where v is the velocity and a is the acceleration

a = ^d⁄_dx(0.5v²) where a is the acceleration, and v is the velocity

∫ a dx = 0.5v²

a + C = 0.5v²

More Information:

Basic Rules with Examples

In Depth Guide to Integrals