Integration

• Integration undoes differentiation — it's the inverse process, formalised by the Fundamental Theorem of Calculus.
• Indefinite integrals always need a "+c" (constant of integration) because constant terms vanish when you differentiate.
• To integrate xⁿ: raise the power by 1, then divide by the new power (never works when n = −1).
• Definite integrals have limits (a to b) — you evaluate the antiderivative at both ends and subtract. The "+c" cancels out, so you don't need it.
• Definite integration gives you the area under a curve — but watch out: area below the x-axis comes out negative, so you take the modulus.
• To find area between a curve and a line, or between two curves, integrate "upper − lower" between the intersection points.
• Special standard integrals exist for sin x, cos x, sec²x, eˣ, and 1/x.
• The Reverse Chain Rule lets you integrate composite functions like (ax+b)ⁿ or e^(ax+b) by "adjusting and compensating."

Think of differentiation and integration as two doors on the same corridor, facing each other. Walk through the "differentiation" door and you turn a function y = f(x) into its gradient function dy/dx. Walk back through the "integration" door and you should land right back where you started... almost.

Here's the catch. Imagine three parallel lines: y = 5x, y = 5x + 2, and y = 5x − 3. They're all different lines, but they all have the exact same gradient: dy/dx = 5. If someone hands you "dy/dx = 5" and asks "what was y?", you genuinely cannot know which of these (or infinitely many others) it was. That's why every indefinite integral ends with + c — a placeholder acknowledging "there could have been a constant here that we lost."

Notation-wise, an integral is written ∫ f(x) dx. Break that symbol down:

• The elongated "S" symbol ∫ literally means "integrate" — it's a stretched S for "Sum," historically tied to summing infinitely many thin slices.
• f(x) is the function you're integrating (called the integrand).
• dx tells you which variable you're integrating with respect to — "do this whole process treating x as the variable."