intermediate

Integration

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Course Syllabus

  1. Introduction to Integration and Indefinite Integrals
  2. Techniques of Integration I: Substitution and Partial Fractions
  3. Techniques of Integration II: Integration by Parts and Trigonometric Integrals
  4. Definite Integrals and the Fundamental Theorem of Calculus
  5. Applications of Integration

Study Notes

Introduction to Integration and Indefinite Integrals

Introduction to Integration and Indefinite Integrals

TL;DR

Integration is like finding the original function when you only know its rate of change. It's essentially the reverse operation of differentiation. Indefinite integrals represent a family of functions that all have the same derivative.

1. The Mental Model

Imagine you know how fast a car is going at every moment, but you want to know how far it's traveled. Integration helps you go from the speed (rate of change) back to the distance (original quantity).

2. The Core Material

When we differentiated a function, we found its derivative, which tells us the slope of the tangent line at any point or the instantaneous rate of change. Integration is this process backwards. If you have a function that's the derivative of another, integration helps you find that original function.

Antiderivatives

A function $F(x)$ is an antiderivative of $f(x)$ if the derivative of $F(x)$ is $f(x)$. In other words, if $F'(x) = f(x)$.

For example, if $f(x) = 2x$, then $F(x) = x^2$ is an antiderivative because the derivative of $x^2$ is $2x$.

But wait, $F(x) = x^2 + 5$ is also an antiderivative, because $\frac{d}{dx}(x^2 + 5) = 2x$.
And $F(x) = x^2 - 100$ is another.
In fact, $F(x) = x^2 + C$ (where C is any constant) is an antiderivative for $f(x) = 2x$.

This constant $C$ is super important! When you differentiate a constant, it becomes zero. So, when you integrate, there's no way to know what that original constant was. We always include a "$+C$" in our answers for indefinite integrals. This $C$ is called the constant of integration.

Indefinite Integrals

The process of finding all antiderivatives of a function is called indefinite integration. We use a special symbol, $\int$, to denote integration.

So, $\int f(x) \,dx$ means "find all antiderivatives of $f(x)$ with respect to $x$."

The result will always be $F(x) + C$, where $F'(x) = f(x)$.

Basic Integration Rules

Here are some fundamental rules that are the reverse of differentiation rules:

  • Power Rule: If you know $\frac{d}{dx}(x^n) = nx^{n-1}$, then $\int x^n \,dx = \frac{x^{n+1}}{n+1} + C$, provided $n
    eq -1$.

    • Example: $\int x^3 \,dx = \frac{x^{3+1}}{3+1} + C = \frac{x^4}{4} + C$.
    • Check: $\frac{d}{dx}(\frac{x^4}{4} + C) = \frac{1}{4} \cdot 4x^3 + 0 = x^3$. It works!

  • Constant Multiple Rule: $\int k \cdot f(x) \,dx = k \cdot \int f(x) \,dx$, where $k$ is a constant.

    • Example: $\i
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