Chapter 4 · Introduction

Chapter 4 — Antiderivatives and Integration

Running differentiation backwards — antiderivatives, areas and the Fundamental Theorem that connects them.

At this point we have seen how to calculate derivatives of many functions and have been introduced to a variety of their applications. We now ask a question that turns this process around: given a function $f$, how do we find a function with the derivative $f$, and why would we be interested in such a function?

The answer leads to antiderivatives and integrals. Given an acceleration we can recover velocity and position; given a rate we can recover a total. And, remarkably, the same antiderivatives compute areas — the content of the Fundamental Theorem of Calculus.

Reverse differentiation→Indefinite integrals→Initial-value problems→Area & FTC→Substitution→Integration by parts
🎯 By the end of this chapter you can…
  • find antiderivatives and indefinite integrals using the basic table and properties;
  • solve initial-value problems, including motion problems;
  • evaluate definite integrals with the Fundamental Theorem of Calculus, and find the average value of a function on an interval;
  • integrate by substitution (the chain rule backwards) and by parts (the product rule backwards).
Teaser — Area as a Limit of Rectangles

Approximate the area under $y=x^2$ from $0$ to $2$ with $n$ rectangles. As $n$ grows the total closes in on exactly $\tfrac83$ — and an antiderivative gets there in one line: $\tfrac{x^3}3\Big|_0^2=\tfrac83$.

4
$n$—
sum of rectangles—
exact area 8/3—
error—
See 4.2 The Definite Integral →

Sections in this chapter

4.1

Antiderivatives

Reverse differentiation, the indefinite integral, the integral table and initial-value problems.

$\int x^n\,dx=\tfrac{x^{n+1}}{n+1}+C$
Open section →4.2

The Definite Integral

Area, properties of definite integrals, both parts of the Fundamental Theorem, and the average value of a function.

$\int_a^bf\,dx=F(b)-F(a)$
Open section →4.3

Integration by Substitution

Spot an inner function and its derivative; rename them u and du.

$\int f(g(x))g^\prime(x)\,dx=\int f(u)\,du$
Open section →4.4

Integration by Parts

Trade a hard integral for an easier one; LIATE; repeated parts.

$\int u\,dv=uv-\int v\,du$
Open section →
✅ Before you start
You must know the derivative table by heart (Formula Sheet) and be quick with exponent laws (0.1 §7) — every integral is a derivative read backwards.
← 3.7 L'Hospital's Rule4.1 Antiderivatives →