Chapter 5 ยท Introduction

Chapter 5 โ€” Techniques of Integration

Calculus I gave you substitution and integration by parts. This chapter completes the toolkit โ€” and shows what to do when no formula exists at all.

At the end of Calculus I you could differentiate almost any formula you were given, but integration was a different story. Substitution undoes the chain rule and integration by parts undoes the product rule โ€” yet integrals as innocent-looking as $\displaystyle\int\sin^2x\,dx$, $\displaystyle\int\sqrt{1-x^2}\,dx$ (a quarter of a circle!) or $\displaystyle\int\frac{dx}{x^2-1}$ still resist both. Engineers meet the first in every alternating-current calculation, the second whenever something is round, and the third in models of chemical reactions and population growth.

This chapter fills those gaps with three algebraic techniques โ€” trigonometric identities, trigonometric substitution and partial fractions โ€” and then faces two honest limitations. Some integrals, like $\displaystyle\int_0^1e^{-x^2}dx$ from statistics, have no antiderivative that can be written with elementary functions, so we learn to compute them numerically to any accuracy we need. And some integrals run over an infinite interval or across a vertical asymptote; with a limit we can still give them a meaning, and often a finite value โ€” the improper integral.

Trigonometric integralsโ†’Trigonometric substitutionโ†’Partial fractionsโ†’Numerical integrationโ†’Improper integrals
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • integrate powers and products of sines, cosines, tangents and secants, using identities and reduction formulas;
  • remove square roots of $a^2-x^2$, $a^2+x^2$ and $x^2-a^2$ by a trigonometric substitution, and use it to find areas of circles, ellipses and segments;
  • split a rational function into partial fractions and integrate each piece;
  • approximate a definite integral with the midpoint, trapezoidal and Simpson's rules, and bound the error;
  • decide whether an improper integral converges, and evaluate it when it does;
  • look at an unfamiliar integral and choose the technique most likely to work.
Teaser โ€” Five Integrals, Five Techniques

Each integral below needs a different section of this chapter. Press โ–ถ Play: for the first four, more and more trapezoids close in on the area (a numerical estimate); for the last, the right end point runs off towards infinity while the area stays below $1$. The exact values come from the techniques you are about to learn โ€” except for the fourth, which has no elementary antiderivative at all.

2
nโ€”
trapezoid estimateโ€”
exact valueโ€”
errorโ€”

Sections in this chapter

5.1

Trigonometric Integrals

Powers of sine and cosine, tangent and secant; half-angle identities, reduction formulas and product-to-sum identities.

$\int\sin^2x\,dx=\tfrac{x}{2}-\tfrac{\sin2x}{4}+C$
Open section โ†’5.2

Trigonometric Substitution

Turn $x$ into the side of a right triangle so the square root disappears; reference triangles, changing limits, completing the square.

$\sqrt{a^2-x^2}\ :\ x=a\sin\theta$
Open section โ†’5.3

Partial Fractions

Split a rational function into simple pieces โ€” linear and quadratic factors, repeated or not โ€” and integrate each one.

$\dfrac{1}{x^2-1}=\dfrac{1/2}{x-1}-\dfrac{1/2}{x+1}$
Open section โ†’5.4

Numerical Integration

Midpoint, trapezoidal and Simpson's rules, with error bounds, for integrals no formula can reach.

$S_n=\tfrac{\Delta x}{3}\left(f_0+4f_1+2f_2+\cdots+4f_{n-1}+f_n\right)$
Open section โ†’5.5

Improper Integrals

Infinite intervals and infinite integrands, handled with limits; convergence, divergence and comparison.

$\int_1^\infty\dfrac{dx}{x^p}$ converges $\iff p\gt1$
Open section โ†’
โœ… Before you start
Every technique in this chapter ends in a substitution or an integration by parts, so make sure both are automatic: review Calculus I, 4.3 Integration by Substitution and 4.4 Integration by Parts. You will also need the trigonometric identities and inverse trigonometric functions (0.1 ยง9 Trigonometry), the Fundamental Theorem of Calculus (4.2), and โ€” for 5.5 โ€” limits at infinity (1.4) and L'Hรดpital's rule (3.7). Not sure where you stand? Take the diagnostic on the Welcome page.
โ† Welcome to Calculus II5.1 Trigonometric Integrals โ†’