Integration Techniques

In the previous chapter, we developed the basic integration formulas and introduced the method of substitution. These tools already handle a wide range of integrals: any polynomial, many rational expressions after a suitable rearrangement, and any integral of the form \int f(g(x))g'(x)\,dx once the right substitution is identified. But they are far from sufficient. Consider, for instance,

$$\int x\sin xdx,\int \ln xdx,\int \frac{x^2+1}{x^3-x}dx,\int \sqrt{1-x^2}dx.$$

None of these yields to the basic formulas or a simple substitution, yet each has a perfectly expressible antiderivative. What is missing is not more formulas but more techniques.

Sections in This Chapter

Section	Technique	What It Handles
6.1	Integrals of Trigonometric Functions	Powers and products of $\sin x$, $\cos x$, $\tan x$, $\sec x$, $\cot x$, and $\csc x$
6.2	Integration by Parts	Products such as $x\sin x$, $x^3{e}^x$, and lone functions such as $\ln x$ and $\arctan x$
6.3	Trigonometric and Hyperbolic Substitutions	Integrands containing $\sqrt{a^2-x^2}$, $\sqrt{a^2+x^2}$, or $\sqrt{x^2-a^2}$
6.4	Integration by Partial Fractions	Every rational function, that is, every ratio of two polynomials

The first section extends our treatment of trigonometric functions, providing systematic methods for integrating expressions involving powers of $\sin x$, $\cos x$, $\tan x$, $\sec x$, and their combinations. The second, integration by parts, is derived from the product rule for differentiation and converts an integral of the form $\int udv$ into one that is often easier to evaluate. The third, trigonometric substitution, handles integrands containing expressions such as $\sqrt{a^2-x^2}$, $\sqrt{a^2+x^2}$, and $\sqrt{x^2-a^2}$ by replacing the variable $x$ with a trigonometric function so as to eliminate the radical. The fourth, partial fractions, provides a method for integrating rational functions by decomposing them into simpler fractions whose antiderivatives are known.

Together, these techniques make it possible to integrate a much larger class of functions than the basic formulas alone would allow.

How the Techniques Fit Together

The four techniques are not four separate islands. They build on one another, and on the substitution rule from the previous chapter:

• Trigonometric substitution (Section 6.3) turns an algebraic integrand into a trigonometric one, at which point the methods of Section 6.1 finish the job.
• The trigonometric integrals $\int \sec xdx$ and $\int {\sec }^{3}xdx$, which appear constantly in trigonometric substitution, are evaluated in Sections 6.1 and 6.2.
• Partial fractions (Section 6.4) reduces any rational function to a sum of simple pieces, but the pieces with irreducible quadratic denominators are then handled by a trigonometric substitution.
• Integration by parts (Section 6.2) is often applied only after a preliminary substitution has simplified the integrand.

For this reason it is worth reading the sections in order, and worth doing the exercises in each section using only the techniques available up to that point.

Where These Techniques Are Used

These are not exercises invented for their own sake. Each technique answers a question that arises repeatedly in science and engineering.

• Mechanical vibrations and alternating current. The average power delivered by an oscillating signal involves $\int {\sin }^2\omega tdt$ and $\int {\cos }^2\omega tdt$, and the interaction between two signals of different frequencies involves $\int \sin ax\cos bxdx$. These are exactly the integrals of Section 6.1.
• Fourier analysis. Decomposing a signal into sines and cosines requires the product-to-sum integrals of Section 6.1, and the coefficients are computed with integration by parts.
• Arc length and surface area. The arc length of a curve involves \int\sqrt{1+\left(f'(x)\right)^{2}}\,dx, which produces radicals of the type $\sqrt{a^2+x^2}$ handled in Section 6.3.
• Areas of circles and ellipses. The integral $\int \sqrt{a^2-x^2}dx$, evaluated by trigonometric substitution, gives the area of a circular segment and, after scaling, of an ellipse.
• Population models and chemical kinetics. The logistic equation and second-order reaction rates lead to integrals of rational functions such as ${\displaystyle \int \frac{dP}{P(K-P)}}$, which are evaluated by partial fractions (Section 6.4).
• Probability and statistics. Expected values of the form $\int xf(x)dx$, where $f$ is a density built from exponentials or powers, are computed by integration by parts.
• Electrostatics and gravitation. The potential due to a charged rod or a thin ring leads to ${\displaystyle \int \frac{dx}{\sqrt{x^2+a^2}}}$, a standard trigonometric or hyperbolic substitution.

A Word on Strategy

Unlike differentiation, integration has no algorithm that always works. Two integrands that look almost identical may need entirely different treatments: $\int x{e}^{x^2}dx$ falls to a one-line substitution, while $\int x{e}^{x}dx$ requires integration by parts, and $\int e^{x^2}dx$ cannot be expressed in terms of elementary functions at all.

What experience gives you is a checklist. Before reaching for a technique, ask:

1. Can the integrand be simplified algebraically or by a trigonometric identity first?
2. Is some part of the integrand the derivative of another part? If so, try a substitution.
3. Does a radical of the form $\sqrt{a^2\pm x^2}$ or $\sqrt{x^2-a^2}$ appear? If so, try a trigonometric or hyperbolic substitution.
4. Is the integrand a product of two functions of different types, one of which becomes simpler when differentiated? If so, try integration by parts.
5. Is the integrand a ratio of polynomials? If so, use partial fractions.

Working through the exercises in each of the following sections is what turns this checklist from a list you read into a reflex you use.