Calculus II: Integration Techniques

Calculus II is where integration stops being a single “anti-derivative rule” and becomes a toolkit. You learn to recognize structure in an integrand, choose an efficient method, and justify convergence when an integral is improper. These skills show up everywhere: computing areas and volumes, modeling work and fluid pressure, and laying groundwork for infinite series.

This article surveys the core integration techniques typically covered in Calculus II: integration by parts, partial fractions, trigonometric substitution, improper integrals, and a first look at series as they relate to integration.

Why integration techniques matter

In Calculus I, many integrals are designed to match a basic rule or a straightforward substitution. In Calculus II, the integrand often blends algebraic, exponential, logarithmic, and trigonometric behavior. The central habit you develop is pattern recognition:

• Is there a product that suggests rearranging via a derivative relationship?
• Is there a rational function that can be decomposed?
• Does a square root like $\sqrt{a^2-x^2}$ hint at a triangle substitution?
• Does the region extend to infinity or approach a vertical asymptote?
• Can a complicated function be represented by a series and integrated term-by-term?

A good method can turn a messy integral into a short sequence of familiar steps.

Integration by parts

Integration by parts is the workhorse for products where substitution does not cleanly apply. It comes from the product rule: \[ (uv)' = u'v + uv' \] Rearranging and integrating yields the formula: \[ \int u\,dv = uv - \int v\,du \]

How to choose \(u\) and \(dv\)

A practical guideline is to choose $u$ so that $du$ becomes simpler, and choose $dv$ so it is easy to integrate to get $v$. Common successful choices include:

• $u=\ln x$ (since $du=\frac{1}{x}dx$ simplifies products)
• $u$ as a polynomial when multiplied by an exponential or trig function
• $u$ as an inverse trig function such as $\arctan x$ or $\arcsin x$

Examples where parts is natural include $\int x{e}^{x}dx$, $\int x\sin xdx$, and $\int \ln xdx$.

Repeated parts and cyclic integrals

Some integrals require applying integration by parts more than once. A classic pattern is products like $e^x\sin x$ or $e^x\cos x$, where applying parts twice brings back the original integral, allowing you to solve algebraically.

Parts also supports reduction formulas, which express $\int {\sin }^{n}xdx$ or $\int {\cos }^{n}xdx$ in terms of lower powers. These are useful when computing areas or when simplifying integrals for later evaluation.

Partial fractions for rational functions

When integrating a rational function $\frac{P(x)}{Q(x)}$, where $P$ and $Q$ are polynomials, partial fraction decomposition is often the fastest route. The goal is to rewrite the fraction as a sum of simpler pieces whose antiderivatives are known (logarithms and arctangents appear frequently).

Step 1: Ensure the fraction is proper

If $\mathrm{deg}P\ge \mathrm{deg}Q$, perform polynomial long division first to rewrite: \[ \frac{P(x)}{Q(x)} = \text{(polynomial)} + \frac{\text{remainder}}{Q(x)} \] Then decompose the proper rational part.

Step 2: Factor the denominator and decompose

The decomposition depends on the factorization of $Q(x)$:

• Distinct linear factors like $(x-a)(x-b)$ lead to terms like $\frac{A}{x-a}+\frac{B}{x-b}$.
• Repeated linear factors like $(x-a{)}^k$ require a full stack: $\frac{A_1}{x-a}+\frac{A_2}{(x-a{)}^2}+\cdots +\frac{A_k}{(x-a{)}^k}$.
• Irreducible quadratics like $x^2+bx+c$ produce terms of the form $\frac{Ax+B}{x^2+bx+c}$.

Integrating these pieces typically yields combinations of $\ln |x-a|$, rational powers, and $\arctan$ terms after completing the square for quadratics.

Trigonometric substitution

Trigonometric substitution targets integrals containing radicals of quadratic expressions. The strategy is to replace $x$ with a trig expression that simplifies the square root using identities like $1-{\sin }^2\theta ={\cos }^2\theta$ and $1+{\tan }^2\theta ={\sec }^2\theta$.

The three standard forms

1. $\sqrt{a^2-x^2}$

Use $x=a\sin \theta$, so $\sqrt{a^2-x^2}=a\cos \theta$.

1. $\sqrt{a^2+x^2}$

Use $x=a\tan \theta$, so $\sqrt{a^2+x^2}=a\sec \theta$.

1. $\sqrt{x^2-a^2}$

Use $x=a\sec \theta$, so $\sqrt{x^2-a^2}=a\tan \theta$.

After substitution, you integrate in $\theta$, then convert back to $x$ using a right-triangle diagram or algebraic identities. This technique is especially common in computing areas, arc lengths, and certain physics applications where square roots of quadratics arise naturally.

Improper integrals

An improper integral is one where the interval is infinite or the integrand becomes unbounded. In these cases, the integral is defined via a limit.

Infinite intervals

For an integral like ${\int }_a^{\infty }f(x)dx$, define: \[ \inta^\infty f(x)\,dx = \lim{b\to\infty}\int_a^b f(x)\,dx \] The key question is whether the limit exists and is finite (convergent) or not (divergent).

Vertical asymptotes

If $f(x)$ blows up at an endpoint or interior point, split and use limits. For example, if $f(x)$ is unbounded at $x=c$ in $(a,b)$, then: \[ \inta^b f(x)\,dx = \inta^c f(x)\,dx + \int_c^b f(x)\,dx \] and each term is evaluated as a limit.

Comparison and \(p\)-tests

Many convergence decisions rely on comparison to standard benchmarks. A central example is: \[ \int1^\infty \frac{1}{x^p}\,dx \] which converges if MATHINLINE50 and diverges if MATHINLINE51_. This links directly to the behavior of tails of distributions and long-range accumulation in applications.

Applications that depend on technique

Integration techniques are not just symbolic exercises. They enable standard Calculus II applications:

• Area between curves often requires setting up integrals with nontrivial algebraic simplification.
• Volumes of solids (disk/washer and shell methods) can lead to polynomials times trig or rational expressions that call for parts or partial fractions.
• Work, fluid force, and moments commonly produce integrals that are not “one-rule” problems and may involve improper behavior when limits extend.

A strong technique choice can be the difference between an integral that is manageable and one that becomes intractable by hand.

Introduction to series through integration

Calculus II typically introduces sequences and series as a new way to represent functions and approximate values. While the full theory of power series often comes later in the course, integration plays an early role.

Term-by-term integration (when allowed)

If a function is represented as a power series on an interval where it converges, you can often integrate term-by-term: \[ \int \left(\sum{n=0}^\infty an x^n\right)\,dx = \sum{n=0}^\infty an \frac{x^{n+1}}{n+1} + C \] This creates new series from old ones and can generate expansions for functions that are difficult to integrate directly.

Practical payoff: approximation

Series provide approximations for definite integrals and function values when closed forms are messy or unavailable. In applied settings, a few terms can yield accurate estimates with clear error behavior once convergence is understood.

Building a reliable method-selection habit

Integration in Calculus II is largely about choosing wisely. A useful workflow is:

1. Simplify algebraically first (factor, cancel, rewrite trig identities).
2. Check for substitution opportunities (especially a derivative hiding inside).
3. If it is a product, consider integration by parts.
4. If it is rational, use partial fractions (after division if needed).
5. If radicals of quadratics appear, consider trigonometric substitution.
6. If bounds are infinite or singularities occur, treat it as improper and test convergence.
7. If a function has a known series form, consider series methods for approximation or simplification.

Mastering these techniques transforms integration from memorization into problem-solving. By the end of Calculus II, you should be able to look at a new