Integrals Cheat Sheet

Key Definitions

• Integral: The mathematical concept of finding the area under a curve.

• Definite Integral: Represents the signed area between the curve and the x-axis over an 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

• Calculates area from x = a to x = b.

• Indefinite Integral: ∫ f(x) dx

• Represents a general antiderivative of f(x).

Fundamental Theorem of Calculus

• Connects differentiation and integration:

• If F is an antiderivative of f, then:

Key Formulas

• Basic Integration Rules:

• ∫ 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 ✅

• Integration by Parts:

• ∫ u dv = uv - ∫ v du ⚠️

Common Mistakes to Avoid

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

Memory Tricks

Tip: "FIND" —

• F: Function to integrate

• I: Identify limits (for definite integrals)

• N: Note the antiderivative

• D: Derive the area or value

Key Diagrams

   |           /
   |         /
 A |       /  Area = ∫_a^b f(x) dx
   |     /
   |   /
   | /
---|----------------
    a       b