Chapter Review

Printed Page 533

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

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)

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)

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)

• 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)

• 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)