Integration Techniques

1. Introduction

Differentiation is mechanical: given a formula, a finite list of rules produces the derivative every time. Integration — finding antiderivatives — is fundamentally harder. There is no universal algorithm, and many perfectly innocent-looking functions (such as $e^{-x^2}$) have no antiderivative expressible in elementary terms. Instead of one method, we build a toolkit of techniques, each one a differentiation rule read backwards, and we learn to recognize the structural cues that tell us which tool to reach for.

The two cornerstones are $u$-substitution, which inverts the chain rule, and integration by parts, which inverts the product rule. Beyond these lie partial fractions for rational functions, trigonometric substitution for expressions with radicals, and a family of trigonometric integral strategies. The Fundamental Theorem of Calculus (FTC) links all of this antidifferentiation back to the computation of definite integrals and areas.

Fluency in integration is largely pattern recognition built through practice. This article lays out each technique with its derivation, a step-by-step procedure, worked examples of increasing difficulty, and the pitfalls that most often trip students.

The Fundamental Theorem of Calculus is the bridge between antidifferentiation and definite integration: Part 1 shows that integration "undoes" differentiation, and Part 2 reduces area computation to evaluating $F(b) - F(a)$. Every technique below is a strategy for finding $F$ when it is not immediately obvious. Unlike differentiation, integration has no algorithm that always terminates — recognizing which tool applies is the central skill.

2. Core Concepts

2.1 Antiderivatives and the Indefinite Integral

$F$ is an antiderivative of $f$ if $F'(x) = f(x)$. Antiderivatives are unique only up to an additive constant: $\int f(x)\,dx = F(x) + C.$ Omitting $C$ on indefinite integrals discards an entire family of solutions.

2.2 The Fundamental Theorem of Calculus

FTC Part 1: If $f$ is continuous on $[a,b]$ and $g(x) = \int_a^x f(t)\,dt$, then $g'(x) = f(x)$.

Proof sketch (Part 1): For small $h > 0$, $g(x+h) - g(x) = \int_x^{x+h} f(t)\,dt$. By continuity, $f(t) \approx f(x)$ on $[x, x+h]$, so the integral $\approx f(x)\,h$, giving $\frac{g(x+h)-g(x)}{h} \approx f(x)$. A similar argument for $h < 0$ completes the proof.

FTC Part 2 (Evaluation): If $F' = f$ on $[a,b]$, then $\int_a^b f(x)\,dx = F(b) - F(a).$

Proof sketch (Part 2): Define $G(x) = \int_a^x f(t)\,dt$. By Part 1, $G' = f$, so $G$ and $F$ are antiderivatives of $f$ and differ by a constant: $G(x) = F(x) + C$. Since $G(a) = 0$, $C = -F(a)$, so $G(b) = F(b) - F(a)$.

2.3 $u$-Substitution Inverts the Chain Rule

The chain rule says $\frac{d}{dx}F(g(x)) = F'(g(x))g'(x)$. Reading backwards: $\int f(g(x))\,g'(x)\,dx = F(g(x)) + C.$ Set $u = g(x)$, $du = g'(x)\,dx$, transforming to $\int f(u)\,du$.

2.4 Integration by Parts Inverts the Product Rule

From $\frac{d}{dx}(uv) = u'v + uv'$, rearrange and integrate: $\int u\,dv = uv - \int v\,du.$ Choose $u$ and $dv$ so $\int v\,du$ is simpler. LIATE orders candidates for $u$: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential.

2.5 Partial Fraction Decomposition

A proper rational function decomposes into simpler fractions. Linear factor $(x-a)$ contributes $\frac{A}{x-a}$; repeated $(x-a)^k$ contributes $\frac{A_1}{x-a} + \cdots + \frac{A_k}{(x-a)^k}$; irreducible quadratic $(x^2+bx+c)$ contributes $\frac{Bx+C}{x^2+bx+c}$. Each piece integrates to logs or arctangents.

2.6 Trigonometric Substitution

Radicals are tamed by Pythagorean identities:

• $\sqrt{a^2 - x^2}$: $x = a\sin\theta$, $dx = a\cos\theta\,d\theta$.
• $\sqrt{a^2 + x^2}$: $x = a\tan\theta$.
• $\sqrt{x^2 - a^2}$: $x = a\sec\theta$.

2.7 Trigonometric Integrals

For $\int \sin^m x\cos^n x\,dx$: if one power is odd, save one factor and convert the rest with $\sin^2 + \cos^2 = 1$; if both even, use half-angle formulas. For $\int \tan^m x\sec^n x\,dx$, similar parity strategies apply.

2.8 Improper Integrals (Preview)

Integrals with infinite limits or integrands blowing up at an endpoint are defined as limits of proper integrals. Convergence depends on comparison with $p$-integrals $\int_1^\infty \frac{1}{x^p}\,dx$ (converges iff $p > 1$).

2.9 Reduction Formulas

Repeated integration by parts yields reduction formulas, e.g. for $I_n = \int \sin^n x\,dx$: $I_n = -\frac{\sin^{n-1}x\cos x}{n} + \frac{n-1}{n}I_{n-2}.$

2.10 Tabular Integration by Parts

For repeated differentiation of an algebraic factor until it vanishes (e.g. $\int x^3 e^x\,dx$), organize $u$ and $dv$ in columns, alternate signs, and multiply diagonally. This streamlines repeated parts without re-labeling $u$ and $dv$ each time.

2.11 Riemann Sums and the FTC Connection

Before antidifferentiating, recall that $\int_a^b f(x)\,dx$ is the limit of $\sum f(x_i^*)\Delta x$. The FTC shows this limit equals $F(b) - F(a)$ whenever $F' = f$, turning area computation into finding an antiderivative.

2.12 Choosing Between Substitution and Parts

If the integrand is $f(g(x))g'(x)$, substitute. If it is a product of an algebraic and a transcendental function, use parts. If it is a rational function, try partial fractions. When in doubt, try substitution first — it is cheaper.