Integration is one of the two fundamental operations of calculus (along with differentiation). At its core, integration is about finding the area under a curve. But it's also much more — it's a powerful tool used across physics, engineering, economics, and many other fields.
An indefinite integral (antiderivative) of a function f(x) is a function F(x) such that F'(x) = f(x). We write it as ∫f(x)dx = F(x) + C, where C is the constant of integration. For example, ∫x² dx = x³/3 + C, because the derivative of x³/3 is x².
A definite integral ∫ₐᵇ f(x)dx represents the signed area between the curve y = f(x) and the x-axis from x = a to x = b. By the Fundamental Theorem of Calculus, this equals F(b) - F(a), where F is any antiderivative of f.
Key integration rules to memorize: the power rule ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (for n ≠ -1), ∫eˣ dx = eˣ + C, ∫sin(x) dx = -cos(x) + C, and ∫cos(x) dx = sin(x) + C. These basic formulas form the building blocks for more complex integrations.
For more complex integrals, you'll need techniques like substitution (u-substitution), integration by parts, partial fractions, and trigonometric substitution. Start with simple problems and gradually work your way up to more challenging ones.