Integrals Cheat Sheet

Key Definitions

• Integral: A mathematical object that represents the area under a curve.

• Definite Integral: Represents the signed area between the curve and the x-axis over a specified interval [a, b].

• Indefinite Integral: Represents a family of functions and includes a constant of integration C.

Important Concepts

Types of Integrals

• Definite Integral: ∫[a, b] f(x) dx gives the area under f(x) from x = a to x = b.

• Indefinite Integral: ∫ f(x) dx = F(x) + C, where F'(x) = f(x).

Fundamental Theorem of Calculus

• Connects differentiation and integration:

• If F is an antiderivative of f, then ∫[a, b] f(x) dx = F(b) - F(a).

Key Formulas

∫ x^n dx = (x^(n+1))/(n+1) + C (for n ≠ -1) ✅
∫ e^x dx = e^x + C ✅
∫ sin(x) dx = -cos(x) + C ✅
∫ cos(x) dx = sin(x) + C ✅
∫ sec^2(x) dx = tan(x) + C ✅
∫ (1/x) dx = ln|x| + C ✅

Common Mistakes to Avoid

Common Mistake: Forgetting to add the constant of integration C in indefinite integrals.

Memory Tricks

Tip: "FIND" — Function, Integrate, New function, Don't forget C.

Key Diagrams

Area under curve:
      |
      |          *
      |        * |
      |      *   |
      |    *     |
      |  *       |
      |*_________|
      a   x   b