Integrals Cheat Sheet

Key Definitions

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

• Definite Integral: Represents the area under the curve between two points, denoted as ∫[a, b] f(x) dx.

• Indefinite Integral: Represents a family of functions, denoted as ∫ f(x) dx = F(x) + C.

Important Concepts

Fundamental Theorem of Calculus

• Connects differentiation and integration.

• States that if F is the antiderivative of f, then:

• ∫[a, b] f(x) dx = F(b) - F(a)

Techniques of Integration

• Substitution: Useful for integrals involving composite functions.

• Integration by Parts: ∫ u dv = uv - ∫ v du

• Partial Fractions: Decomposing rational functions into simpler fractions.

Key Formulas

• Basic 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

• Definite Integral:

• ∫[a, b] f(x) dx = F(b) - F(a)

Common Mistakes to Avoid

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

Memory Tricks

Tip: "SIMPLE" —

• S: Substitution

• I: Integration by Parts

• M: Method of Partial Fractions

• P: Properties of Definite Integrals

• L: Limits in Definite Integrals

• E: Evaluate the Integral

Key Diagrams

Area under curve:
   |\
   | \
   |  \
   |   \
   |    \   f(x)
   |     \
   |------\________
   a      b

Remember to visualize the area under the curve for definite integrals!