Techniques of Integration

Chapter Review

533

THINGS TO KNOW

7.1 Integration by Parts

Integration by parts formula \(\int u dv=uv-\int v\,du\) (p. 472)
Guidelines for choosing \(u\) and \(dv\) (Table 1, p. 475)

7.2 Integrals Containing Trigonometric Functions

Procedures:

Trigonometric integrals of the form \(\int \sin ^{n}x\,dx\) or \(\int \cos ^{n}x\,dx,\) \(n \ge 2\) an integer (pp. 480, 481)
Trigonometric integrals of the form \(\int \sin ^{m}x\,\cos ^{n}x\,dx\) (p. 483)
Trigonometric integrals of the form \(\int \tan ^{m}x\,\sec ^{n}x\,dx\) or \(\int \cot ^{m}x\,\csc ^{n}x\,dx\) (pp. 483-485)
Trigonometric integrals of the form \(\int \sin (ax) \,\sin (bx) \,dx\), \(\int \sin (ax) \cos (bx)\,dx,\) or \(\int \cos (ax) \,\cos (bx) \,dx\) (p. 485)

7.3 Integration Using Trigonometric Substitution: Integrands Containing \(\sqrt{a^{2}-x^{2}}\), \(\sqrt{x^{2}+a^{2}}\), or \(\sqrt{x^{2}-a^{2}}\)

See Table 2 (p. 488):

Integrals containing \(\sqrt{a^{2}-x^{2}}\): Use the substitution \( x=a \sin \theta, -\frac{\pi }{2} \le \theta \le \frac{\pi }{2}\). (p. 488)
Integrals containing \(\sqrt{a^{2}+x^{2}}\): Use the substitution \( x=a \tan \theta, -\frac{\pi }{2} \lt \theta \lt \frac{\pi }{2}\). (p. 489)
Integrals containing \(\sqrt{x^{2}-a^{2}}\): Use the substitution \( x=a \sec \theta, 0 \le \theta \lt \frac{\pi }{2}\) or \(\pi \le \theta \lt \) \( \frac{3\pi }{2}\). (p. 491)

7.4 Substitution: Integrands Containing \(ax^{2}+bx+c\) (p. 496)

7.5 Integration of Rational Functions Using Partial Fractions

Definitions:

Proper and improper rational functions (p. 499)
Partial fractions (p. 499)
Irreducible quadratic factor (p. 503)

Partial Fraction Decomposition of \(R(x) =\frac{p(x)}{q(x)}, q(x) ≠ 0\):

Case 1: a proper rational function whose denominator contains only distinct linear factors (p. 500)
Case 2: a proper rational function whose denominator contains a repeated linear factor \((x-a) ^{n}, n \geq 2\), (p. 502)
Case 3: a proper rational function whose denominator contains a distinct irreducible quadratic factor \(x^{2}+bx+c\). (p. 503)
Case 4: a proper rational function whose denominator contains a repeated irreducible quadratic factor \((x^{2}+bx+c)^{n}, n \geq 2\) an integer, (p. 504)

7.6 Integration Using Numerical Techniques

Trapezoidal Rule: (p. 509)
Error in the Trapezoidal Rule \(\le \frac{(b-a)^{3}M}{12n^{2}},\) where \(M\) is the absolute maximum value of \(\vert f^{\prime \prime} \vert\) on the closed interval \([a,b]\). (p. 512)
Simpson’s Rule: (p. 515)
Error in Simpson’s Rule \(\le \frac{(b-a)^{5}M}{180n^{4}},\) where \( M\) is the absolute maximum value of \(\vert f^{\,(4)}(x)\vert\) on the closed interval \([a,b]\). (p. 516)

7.7 Integration Using Tables and Computer Algebra Systems (p. xx)

7.8 Improper Integrals

Improper integrals of the form \(\int_{a}^{\infty }f(x)\,dx; \int_{-\infty }^{b}f(x)\,dx;\) and \(\int_{-\infty }^{\infty }f(x)\,dx\) (pp. 524, 525)
Converge; diverge (p. 524)
Improper integrals \(\int_{a}^{b} f(x)\,dx\), where \(f(a) ,\) \(f( b)\), or \(f( c)\) is not defined, \(a \lt c \lt b.\) (pp. 527 and 528)
Theorem \(\int^{\infty}_1 \frac{dx}{x^p}\) converges if \(p>1\) and diverges if \(p \le 1\). (p. 526)

Comparison Test for Improper Integrals (p. 529)

534

OBJECTIVES

Section	You should be able to…	Examples	Review Exercises
7.1	1 Integrate by parts (p. 472)	1–6	8, 9, 16, 19, 21, 25, 47, 48
	2 Derive a formula using integration by parts (p. 476)	7, 8	37, 38
7.2	1 Find integrals of the form \(\int \sin ^{n}x\,dx\) or \(\int \cos ^{n}\!x\,dx,\) \(n \ge 2\) an integer (p. 480)	1–3	5, 31
	2 Find integrals of the form \(\int \sin ^{m}x\,\cos^{n}x\,dx\) (p. 483)	4	10, 28
	3 Find integrals of the form \(\int \tan ^{m}x\,\sec^{n}x\,dx\) or \(\int \cot ^{m}x\,\csc ^{n}x\,dx\) (p. 483)	5–7	3, 4
	4 Find integrals of the form \(\int \sin (ax) \sin (bx) \,dx\), \( \int \sin (ax) \,\cos (bx)\,dx,\) or \(\int \cos (ax) \,\cos (bx) \,dx\) (p. 485)	8	32, 33
7.3	1 Find integrals containing \(\sqrt{a^{2}-x^{2}}\) (p. 488)	1	6, 11
	2 Find integrals containing \(\sqrt{x^{2}+a^{2}}\) (p. 489)	2, 3	18, 24, 26
	3 Find integrals containing \(\sqrt{x^{2}-a^{2}}\) (p. 491)	4	7, 30
	4 Use trigonometric substitution to find definite integrals (p. 492)	5, 6	34, 35
7.4	1 Find an integral that contains a quadratic expression (p. 496)	1–3	1, 14, 20
7.5	1 Integrate a rational function whose denominator contains only distinct linear factors (p. 500)	1, 2	12, 27, 29
	2 Integrate a rational function whose denominator contains a repeated linear factor (p. 502)	3	17, 23
	3 Integrate a rational function whose denominator contains a distinct irreducible quadratic factor (p. 503)	4	2, 15
	4 Integrate a rational function whose denominator contains a repeated irreducible quadratic factor (p. 504)	5	22
7.6	1 Approximate an integral using the Trapezoidal Rule (p. 508)	1–5	49(a), 50
	2 Approximate an integral using Simpson’s Rule (p. 514)	6–8	49(b)
7.7	1 Use a Table of Integrals (p. 520)	1–3	36(a)
	2 Use a computer algebra system (p. 522)	4	36(b)
7.8	1 Find integrals with an infinite limit of integration (p. 524)	1, 2	39, 42, 44
	2 Interpret an improper integral geometrically (p. 525)	3, 4	51, 52
	3 Integrate functions over \([a,b]\) that are not defined at an endpoint (p. 527)	5–7	40, 41, 43
	4 Use the Comparison Test for Improper Integrals (p. 529)	8	45, 46