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THE METHOD OF PARTIAL FRACTIONS
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and (see Problem 40.4) sin 2 = so that we have, nally, 1 1 x dx 1 = ln |x + 1| + In (x 2 + 2x + 2) 2 + 2x + 2)2 2 + 2x + 2 2 2 x (x + 1)(x 1 x+1 1 (x + 1) + 1 +C + tan 2 2 x 2 + 2x + 2 1 x = + tan 1 (x + 1) ln |x + 1| + C ln (x 2 + 2x + 2) + 2 2 x + 2x + 2 2(x + 1) 2 tan = 2 x + 2x + 2 1 + tan2
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Supplementary Problems
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40.6 Find the following antiderivatives: dx (a) 2 9 x (d) (g) ( j) (m) (p) (s) 2x 2 + 1 dx (x 1)(x 2)(x 3) x dx x 4 13x 2 + 36 x+4 x 3 + 6x 2 + 9x x 2 dx dx x(x 2 + 1)2 x 1 x 3 + 2x 2 x 2 dx dx (b) (e) (h) (k) (n) (p) (t) x dx (x + 2)(x + 3) x2 4 dx x 3 3x 2 x + 3 x 5 dx x 2 (x + 1) x 4 dx x 3 2x 2 7x 4 dx (x 2 + 1)(x 2 + 4) x 2 dx (x 1)(x 2 + 4)2 x2 + 2 x(x 2 + 5x + 6) (c) (f ) (i) (l) (o) (r) (u) x 4 4x 2 + x + 1 dx x2 4 x3 + 1 dx x(x + 3)(x + 2)(x 1) 2x dx (x 2)2 (x + 2) dx x(x 2 + 5) x4 + 1 dx x 3 + 9x x3 + 1 x(x 2 + x + 1)2 dx 1 + ex dx
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(x 1)(x 2 + 4x + 5)
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40.7 Show that p(x) = x 2 + bx + c is irreducible if and only if c (b2 /4) > 0. [Hint: A quadratic polynomial is irreducible if and only if it has no linear factor; that is (by Theorem 7.2), if and only if it has no real root.] 40.8 (a) Find the area of the region in the rst quadrant under the curve y = l/(x 3 + 27) and to the left of the line x = 3. (b) Find the volume of the solid generated by revolving the region of part (a) around the y-axis. 40.9 Find dx . 1 sin x [Hint: See Problem 40.4.]
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40.10 Find
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cos x dx . sin x 1 equivalent.
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(a) Use the method of Problem 40.4. (b) Use the Quick Formula II. (c) Verify that your answers are
40.11 Evaluate the following integrals involving fractional powers: dx dx dx (a) (c) (b) 3 x x x 1 + 3x 1+ 4 x 1 (e) dx x+1 (f ) 1 + ex dx (g) x 2/3 dx x+1
dx 3 x+ x
[Hints: Part (a) let x = z3 ; part (b) let x 1 = z4 ; part (c) let 1 + 3x = z2 ; part (d) let x = z6 .]
APPENDIX A
Trigonometric Formulas
cos2 + sin2 = 1 sin x 1 = tan x = cos ( + 2 ) = cos cos x cot x 1 cos x sin ( + 2 ) = sin = cot x = cos ( ) = cos sin x tan x 1 sin ( ) = sin sec x = cos (u + v) = cos u cos v sin u sin v cos x 1 cos (u v) = cos u cos v + sin u sin v csc x = sin (u + v) = sin u cos v + cos u sin v sin x sin (u v) = sin u cos v cos u sin v tan ( x) = tan x sin 2 = 2 sin cos tan (x + ) = tan x cos 2 = cos2 sin2 1 + tan2 x = sec2 x 1 + cot 2 x = csc2 x = 2 cos2 1 = 1 2 sin2 1 + cos tan u + tan v tan (u + v) = cos2 = 2 2 1 tan u tan v 1 cos tan u tan v 2 = tan (u v) = sin 2 2 1 + tan u tan v cos = sin ; sin ( ) = sin ; sin ( + ) = sin 2 = cos ; cos ( ) = cos ; cos ( + ) = cos sin 2 Law of cosines: c2 = a2 + b2 2ab cos sin B sin C sin A = = Law of sines: a b c
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