The Integration
Master Reference

Before we prove anything, here's the movie in one sentence: if you keep track of a running total \(F(x) = \int_a^x f(t)\,dt\), then the rate at which that total changes is exactly \(f(x)\). That's it. That's the whole theorem.

Part I — The Derivative of an Accumulation Function

Define the accumulation function:

What does this mean? We're sweeping from \(a\) to \(x\) and adding up all the tiny \(f(t)\,dt\) contributions. \(F(x)\) is the running total. Now ask: what's \(F'(x)\)?

Part II — The Evaluation Theorem

This is the one you actually use to compute integrals. If \(F'(x) = f(x)\), then:

An antiderivative \(F(x)\) of \(f(x)\) is any function satisfying \(F'(x) = f(x)\). We always add \(+C\) because the derivative of a constant is zero — you can't recover the constant from the derivative alone.

Power Rule (Reversed)

Core Antiderivative Table

Layer 1 — Single Variable: \(dx\) as a tiny input change

If \(y = f(x)\), the differential \(dy\) is defined as:

Mechanically: \(dx\) is an infinitesimally small change in the input. \(dy\) is the corresponding linear approximation to the change in output. This is not the actual change \(\Delta y\); it's the change predicted by the tangent line.

Layer 2 — Substitution: how differentials transform

If \(u = g(x)\), then differentiating both sides:

This is not a trick — it's the chain rule applied to the differential. When we "substitute," we're literally replacing the \(g'(x)\,dx\) piece inside the integral with \(du\). The differential is the bridge.

Layer 3 — Multivariable: Total Differential

If \(z = f(x, y)\), the total differential captures how \(z\) changes due to changes in both \(x\) and \(y\):

Each term \(\frac{\partial f}{\partial x}\,dx\) says: "how much does \(z\) change if only \(x\) wiggles by \(dx\)?" The total differential adds these contributions. This is the multivariable tangent plane approximation:

The Grand Payoff — Area Elements

When you change coordinates, the differential area element \(dA\) transforms too. This is why:

• Polar: \(dA = r\,dr\,d\theta\) (the \(r\) comes from stretching)
• Cylindrical: \(dV = r\,dz\,dr\,d\theta\)
• Spherical: \(dV = \rho^2\sin\phi\,d\rho\,d\theta\,d\phi\)

Those scale factors aren't magic — they're the determinants of the Jacobian matrix of the coordinate transformation. More on this in Section 10.

The Mechanism — Chain Rule Reversed

The chain rule says: \(\frac{d}{dx}[F(g(x))] = F'(g(x))\cdot g'(x)\). So integrating both sides:

Now set \(u = g(x)\), so \(du = g'(x)\,dx\). The integral becomes:

That's it. The \(g'(x)\,dx\) piece in the original integrand must be present (up to a constant) for u-sub to work. You're recognizing that the derivative of the inner function is sitting there, waiting.

Algorithm

Definite Integral Bound Change

When limits are given, transform them: if \(u = g(x)\), then the limits \(x = a, b\) become \(u = g(a), g(b)\). No need to back-substitute.

Proof from the Product Rule

How to Choose \(u\) and \(dv\) — LIATE Guide

Set \(u\) = the function that comes first in this priority list (it gets differentiated, so you want it to simplify):

LIATE is a heuristic, not a law. The goal: pick \(u\) that simplifies when differentiated, and \(dv\) that you can actually integrate.

Repeated IBP

Sometimes you apply IBP twice (or more). In cases like \(\int x^2 e^x\,dx\), each round reduces the power of \(x\) by one. A tabular method organizes this:

Multiply diagonally with alternating signs: \(\int x^2 e^x\,dx = x^2 e^x - 2x e^x + 2e^x + C = e^x(x^2 - 2x + 2)+C\).

Essential Pythagorean Identities (always keep these hot)

Family 1

Family 2

The Three Core Substitutions

Why the Identities Kill the Radical

For \(x = a\sin\theta\):

When to Use

When the integrand is a rational function \(\frac{P(x)}{Q(x)}\). First check: if degree of \(P \geq\) degree of \(Q\), do polynomial long division first to get \(\frac{P}{Q} = (\text{polynomial}) + \frac{R(x)}{Q(x)}\) where \(\deg R < \deg Q\).

The Four Cases

Case 1 — Distinct Linear Factors

If \(Q(x) = (x-a)(x-b)\cdots\), write:

Multiply both sides by \((x-a)(x-b)\), then plug in \(x = a\) and \(x = b\) to solve for \(A\) and \(B\) directly (the "cover-up method").

Case 2 — Repeated Linear Factors

For \((x-a)^k\), you need one term for each power:

Case 3 — Irreducible Quadratics

For a factor \(x^2 + bx + c\) with no real roots:

The numerator needs to be linear (one degree below the denominator).

Case 4 — Repeated Irreducible Quadratics

Type I — Infinite Limits

For doubly infinite integrals, split at any convenient point \(c\): both halves must converge independently.

Type II — Vertical Asymptote

If \(f\) blows up at \(x = c \in [a,b]\):

Both limits must converge. A common mistake: forgetting the singularity in the middle and computing as if the function were bounded.

1D Case — Sanity Check

You already know this: if \(u = g(x)\), then \(du = g'(x)\,dx\). The "Jacobian" in 1D is just \(g'(x)\). It tells you how the length element stretches.

2D Jacobian — The Core Construction

Suppose we change from \((x,y)\) to \((u,v)\) via \(x = x(u,v)\), \(y = y(u,v)\). A tiny rectangle \(du\,dv\) in \((u,v)\)-space maps to a parallelogram in \((x,y)\)-space. The area of that parallelogram is:

where the Jacobian determinant is:

Polar Coordinates — Full Derivation

Let \(x = r\cos\theta\), \(y = r\sin\theta\). Compute the Jacobian:

General Change of Variables Formula

Fubini's Theorem — Iterated Integration

For a rectangle \(R = [a,b]\times[c,d]\):

You can switch the order of integration for a rectangle (as long as \(f\) is continuous). The intuition: you're summing along columns then rows, or rows then columns — same total.

Non-Rectangular Regions — Type I and Type II

Type I (bounded by functions of \(x\)): \(R = \{(x,y): a \leq x \leq b,\; g_1(x) \leq y \leq g_2(x)\}\):

Type II (bounded by functions of \(y\)): integrate in the other order.

Geometric Meaning

Average Value

Polar (2D)

Use when region is a disk, annulus, wedge, or any boundary described by \(r = f(\theta)\).

Cylindrical (3D)

Use for cylinders, cones, any solid with circular cross-sections. The Jacobian is the same \(r\) as polar — the \(z\) direction doesn't get distorted.

Spherical (3D) — Full Jacobian Derivation

Variables: \(\rho\) = radial distance from origin, \(\phi\) = polar angle from \(+z\)-axis (\(0 \leq \phi \leq \pi\)), \(\theta\) = azimuthal angle in \(xy\)-plane.

Line Integrals — Scalar Field

Integrate \(f(x,y,z)\) along a curve \(C\) parametrized by \(\textbf{r}(t) = (x(t), y(t), z(t))\) for \(t \in [a,b]\):

The \(ds = |\textbf{r}'(t)|\,dt\) is the arc-length element — the tiny piece of path length. Geometrically this is the "curtain area" under \(f\) draped along \(C\).

Line Integrals — Vector Field (Work)

If \(\textbf{F}(x,y,z)\) is a vector field (e.g., a force), the work done along \(C\) is:

Here \(d\textbf{r} = \textbf{r}'(t)\,dt\) is a tiny displacement vector along the path. The dot product picks out the component of \(\textbf{F}\) in the direction of motion — only that component does work.

Surface Integrals — Scalar Field (Surface Area)

For a surface parametrized by \(\textbf{r}(u,v)\) over a region \(D\):

The cross product \(\textbf{r}_u \times \textbf{r}_v\) gives a vector perpendicular to the surface, and its magnitude is the area of the tiny parallelogram patch. When \(f=1\), this is just the surface area.

Surface Integrals — Vector Field (Flux)

The flux of \(\textbf{F}\) through surface \(S\) measures how much \(\textbf{F}\) flows through \(S\):

Here \(\hat{\textbf{n}}\) is the unit normal to the surface and \(d\textbf{S} = \hat{\textbf{n}}\,dS\). Only the component of \(\textbf{F}\) perpendicular to the surface contributes to flux.

When a function has no closed-form antiderivative (e.g., \(e^{-x^2}\)), we approximate using simple shapes.