Introduction
Precalculus
Getting Started
Calculus 1
1. Limits
3. Derivatives
5. Applications of Derivatives
6. 1. Increasing and Decreasing Function
  2. Minimum and Maximum Values
7. Integrals
8. 1. Indefinite Integrals
  2. Computing Indefinite Integrals
  3. Substitution Rule for Indefinite Integrals
  4. Area Problem
Calculus 2
1. Integration Techniques
2. 1. Integration by Parts
  2. Integrals Involving Trig Functions
  3. Trig Substitutions
  4. Partial Fractions
  5. Integrals Involving Roots
  6. Integrals Involving Quadratics
3. Parametric Equations and Polar Coordinates
4. 1. Differentiation Formulas
  2. Product and Quotient Rule
  3. Chain Rule
  4. Implicit Differentiation
  5. Higher Order Derivatives
  6. Logarithmic Differentiation
5. Series and Sequences
6. 1. Indefinite Integrals
  2. Computing Indefinite Integrals
  3. Substitution Rule for Indefinite Integrals
  4. Area Problem
7. Vectors
8. 3-Dimensional Space
Calculus 3
1. LimitThree Dimensional Coordinate Systems
2. 1. Equations of Lines
  2. Equations of Planes
  3. Quadratic Surfaces
  4. Functions of Multiple Variables
3. Partial Derivatives
4. 1. Higher Order Partial Derivatives
  2. Differentials
  3. Chain Rule
  4. DirectioChain Rulenal Derivatives
  5. Gradient
5. Multiple Integrals
6. 1. Iterated Integrals in Cylindrical Coordinates
  2. Double Integrals
  3. Double Integrals in Polar Coordinates
  4. Triple Integrals in Spherical Coordinates
  5. Change of Variables
  6. Surface Area
7. Line Integrals
8. 1. Line Integrals With Respect to Arc Length
  2. Double Integrals in Polar Coordinates
  3. Line Integrals With Respect to x and y
  4. Line Integrals of Vector Fields
  5. Fundamental Theorem of Line Integrals
  6. Conservative Vector Fields
  7. Potential Functions
  8. Green's Theorem
9. Surface Integrals
10. 1. Parametric Surfaces
  2. Surface Integrals
  3. Surface Integrals of Vector Fields
  4. Stokes' Theorem
  5. Divergence Theorem
Cheatsheet

Integrals

What is Integrals?

Integral calculus helps in finding the anti-derivatives of a function. These anti-derivatives are also called the integrals of the function. The process of finding the anti-derivative of a function is called integration. The inverse process of finding derivatives is finding the integrals. The integral of a function represents a family of curves. Finding both derivatives and integrals form the fundamental calculus.

Given a function, \(f\left( x \right)\), an anti-derivative of \(f\left( x \right)\) is any function \(F\left( x \right)\) such that $$F'\left( x \right) = f\left( x \right)$$ If \(F\left( x \right)\) is any anti-derivative of \(f\left( x \right)\) then the most general anti-derivative of \(f\left( x \right)\) is called an indefinite integral and denoted, $$\int{{f\left( x \right),dx}} = F\left( x \right) + c\hspace{0.25in},c{\mbox{ is an arbitrary constant}}$$ In this definition the \(\int{{}}\) is called the integral symbol, \(f\left( x \right)\) is called the integrand, \(x\) is called the integration variable and the "\(c\)" is called the constant of integration.

Example of Integral

$$ \int \sin^3(x) e^{-5x} dx $$

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