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1 Definitions, Identities, Basic Ops 1.1 Exponents and Radicals 1.1.1 Identities and Formulas 1. x0 = 1 2. x1 = x 3. xm xn = xm+n 4. xm xn = xm+n m 5. x n = xm−n x 6. (xm )n = xmn 2.2 2. 3. 4. x−n = 1n x 8. (xy)n = xn y n n 9. ( x )n = xn y y √ 1/n n 10. x = x √ √ n 11. xm/n = √ xm = ( n x)m √ √ 12. n xy = n x n y √ q nx √ 13. n x = n y y 7. 5. 6. 7. 8. 9. 10. 1.2 Logarithms and Exponentials 1.2.1 Definition of logarithm y loga x = y ⇐⇒ a = x 11. 12. 1.2.2 Identities and Formulas 1. loga 0 = −∞ 2. loga 1 = 0 3. loga a = 1 4. loge x = ln x 5. loga ax = x 6. ln ex = x 1 7. loga x = x = logx a 8. loga xy = loga x + loga y 9. loga x = loga x − loga y y 10. loga xc = c loga x ln x 11. e = x 12. ax+y = ax ay x 13. ax−y = ay a y 14. ax = axy 15. abx = ax bx 1.3 Trigonometry 1.3.1 Identities and Formulas 1. sin θ = 1/ csc θ 2. cos θ = 1/ sec θ 3. tan θ = sin θ/ cos θ 4. cot θ = cos θ/ sin θ = 1/ tan θ 5. sec θ = 1/ cos θ 6. csc θ = 1/ sin θ 2 2 7. sin θ + cos θ = 1 2 2 8. tan θ + 1 = sec θ 2 2 9. cot θ + 1 = csc θ 10. sin −θ = − sin θ 11. cos −θ = − cos θ 12. tan −θ = − tan θ π − θ) = cos θ 13. sin( 2 π 14. cos( − θ) = sin θ 2 π 15. tan( − θ) = cot θ 2 1.3.2 1. 2. 3. 4. Addition and Subtraction sin(x + y) = sin x cos y + sin y cos x sin(x − y) = sin x cos y − sin y cos x cos(x + y) = cos x cos y − sin x sin y cos(x − y) = cos x cos y + sin x sin y tan x + tan y 5. tan(x + y) = 1 − tan x tan y tan x − tan y 6. tan(x − y) = 1 + tan x tan y 1.3.3 Double-Angle Formulas sin 2x = 2 sin x cos x 2 2 2 cos 2x = cos x − sin x = 2 cos x − 1 2 = 1 − 2 sin x tan 2x = 2 tan x 1 − tan2 x 1.3.4 Half-Angle Formulas 2 sin x = (1 − cos2x)/2 2 2. cos x = (1 + cos2x)/2 13. 14. Z Z Z Z xn+1 n x dx = + C, (n 6= −1) n+1 1 dx = ln |x| + C x x x e dx = e + C ax x a dx = +C ln a sin xdx = − cos x + C Z tan xdx = ln | sec x| + C Z Z Z Z csc x cot xdx = − csc x + C Z 15. 16. Z Z sin x sin y = 1/2[cos (x − y) − cos (x + y)] = x3 − x2 − x + 1 x4 − 2x2 + 4x + 1 −x 2 4x cosh xdx = sinh x + C x3 + 4x + 2x2 − x + 4 x(x2 + 4) 2x 2x 2 2 Z 2π q r 2 (1 − cos θ)2 + r 2 sin2 θdθ 0 Z 2π q = r 2 (1 − 2 cos θ + cos2 θ + sin2 θ)dθ 0 Z 2π q = r 2(1 − cos θ)dθ 0 1 2 sin x = (1 − cos2x), θ = 2x, 2 2 1 − cos θ = 2 sin (θ/2) q q 2(1 − cos θ) = 4 sin2 (θ/2) C + 2x − 1 x+2 = 2| sin (θ/2)| = 2sin(θ/2) Z 2π 2π sin (θ/2)dθ = 2r[−2 cos (θ/2)]0 0 = 2r dx = 2r[2 + 2] = 8r 4x x3 − x2 − x + 1 3.1.3 A = + x−1 B (x − 1)2 + C x + 1S = Q(x) contains irreducible quadratic factors dx 1 tan−1 ( x ) + C = a a x2 +a2 Z 2x2 − x + 4 R = Arc Length L = 2 − x + 1 = (x − 1) (x + 1) x3 − x2 − x + 1 R B + = x+1+ x3 − x2 − x + 1 Use A x Z x4 − 2x2 + 4x + 1 2.6.3 3.1.2 Q(x) is a product of linear factors, some of which are repeated 2.6.2 3 ) dx x(2x − 1)(x + 2) x 2 +x+2+ + 2x + 2 ln |x − 1| + C 2 2x3 + 3x2 − 2x sinh xdx = cosh x + C Surface Area Z 2π 2πr(1 − cos θ) 0 q r 2 (1 − cos θ)2 + r 2 sin2 θdθ = 4πr = A + x − x + 4 = A(x 2 Bx + C x2 + 4 + 4) + (Bx + C)x − x + 4 = (A + B)x 2 = + Cx + 4A (A + B = 2), (C = −1), (4A = 4) Z 2π 0 (1 − cos θ)(sin θ )dθ 2 Z 2π θ sin cos θdθ 0 2 Z θ 1 2π 2 2 4πr [−2 cos ]0 π − sin (3θ/2)dθ 2 2 0 1 Z 2π θ + sin dθ 2 0 2 θ 1 θ 2π 2 4πr [−2 cos + cos (3θ/2) − cos ]0 2 3 2 1 2 cos 3π) 4πr [(−3 cos π + 3 1 − (−3 cos 0 + cos 0)] 3 1 1 2 2 16 4πr [3 − + 3 − ] = 4πr [ ] 3 3 3 64 2 πr 3 = 4πr dx 2 2 Z 2π 0 (sin θ 2 dθ − = Integration by parts that seem to flip-flop R x A = 1, B = 1, C = −1 (e.g. e sin xdx) can actually be solved this Z way since you will eventually end up with the Z 2x2 − x + 4 1 x−1 original eqn., then add it to the left side and dx = ( + )dx x3 + 4x x x2 + 4 divide right side by 2. Z Z Z 2.4 Trigonometric Integrals = R x−1 x 1 2.4.1 sinm x cosn xdx dx = ( dx − )dx x2 + 4 x2 + 4 x2 + 4 1. m is odd, save a sin x and use = sin2 x = 1 − cos2 x, u = cos x 2. n is odd, save a cos x and use 2.6.4 Q(x) contains a repeated irrecos2 x = 1 − sin2 x, u = sin x ducible quadratic factor 3. m and n both odd, use either of the above 4 Polar Coordinates 4. m and n both even, use half-angle Z 1 − x + 2x2 − x3 dx identities or sin x cos x = 1 sin 2x 2 x(x2 + 1)2 R 2.4.2 tanm x secn xdx 1 − x + 2x2 − x3 A Bx + C Dx + E syntax:(r, θ) = + + 1. m is odd, save a sec x tan x and use x(x2 + 1)2 x x2 + 1 (x2 + 1)2 polar to cartesian:x = r cos θ, y = r sin θ tan2 x = sec2 x − 1, u = sec x y 2 2 2 2. n is even, save a sec2 x and use cartesian to polar:r = x + y , tan θ = x sec2 x = 1 + tan2 x, u = tan x as parametric x:x = r cos θ = f (θ) cos θ 2.7 Arc Length 2.5 Trigonometric Substitution q Rb q as parametric y:x = r sin θ = f (θ) sin θ ′ (x)]2 dx • 1 + [f 2 2 1. a − x , sub x = a sin θ, use 1 − a dy Rd q dr sin θ + r cos θ dy 2 2 1 + [f ′ (y)]2 dy • sin θ = cos θ q c derivative: = dθ = dθ dx dr cos θ − r sin θ Rx q dx a2 + x2 , sub x = a tan θ, use 1 + 2. • s(x) = a 1 + [f ′ (t)]2 dt dθ dθ 2 θ = sec2 θ tan q tangent: calc derivative @ given θ Surface Area x2 − a2 , sub x = a sec θ, use 2.8 3. or htan/vtan from parametric Z b q sec2 θ − 1 = tan2 θ Z b 1 y = f (x)abt the x: 2πy 1 + [f ′ (x)]2 dx 2 q A = r dθ a a 2 R 9−x2 Z d q dx Ex: s 2 Z ′ 2 x x = f (y)abt the x: 2πy 1 + [f (y)] dy dr b c )2 dθ r2 + ( L = 1. x = 3 sin θ, dx = 3 cos θdθ, sin θ = Z b a q dθ x , θ = sin−1 x ′ 2 y = f (x)abt the y: 2πx 1 + [f (x)] dx 3 3 q q a = 2. 9 − x2 = 9 − 9 sin2 θ Z d q 4.1 Equation conversion to Cartesian q p x = f (y)abt the y: 2πx 1 + [f ′ (y)]2 dy 2 2 9(1 − sin θ) = 9 cos θ = 3 cos θ c Use sin θ = y/r, cos θ = x/r, r 2 = x2 + y 2 Z q 9 − x2 x2 dx = = cos x cos y = 1/2[cos (x + y) + cos (x − y)] + 2 x−1 x2 x2 + 2x − 1 1 −1 dx = tan x+C x2 + 1 1 −1 dx = sin x+C q 1 − x2 2.3 Integration by Parts 2.3.1 Definition Z Z udv = uv − vdu (x x2 + 2x − 1 Z sec x tan xdx = sec x + C dx = 1 ln | x−a | + C 2a x+a x2 −a2 R dx 19. = 1 tan−1 ( x ) + C a a x2 +a2 R 1 [sec x tan x 20. sec3 xdx = 2 ln | sec x + tan x|] + C 18. x3 Z Q(x) is a product of distinct linear factors 2.6.1 2 sec xdx = tan x + C 2 csc xdx = − cot x + C dx = 3 sec xdx = ln | sec x + tan x| + C Z Z = cos xdx = sin x + C Product Formulas sin x cos y = 1/2[sin (x + y) + sin (x − y)] x−1 Z Z 17. 1. 1.3.5 Z x3 + x Indefinite Integrals Z 1. kdx = kx + C Z 3 cos θ 9 sin2 θ Z 9 cos2 θ 9 sin2 θ 3 cos θdθ dθ = Z 2 cot θdθ = Z 2 [csc θ − 1]dθ 3 r = 3 sin θ Parametric Equations dy/dt dx dy , = 6= 0 • dx dx/dt dt • x = f (t), y = g(t), α 6 t 6 β dy • h-tan: = 0& dx 6= 0 dt dt dy • v-tan: 6= 0& dx = 0 dt dt • h/v-tan points: find d(y/x)/dt = 0 @ t=z, plug t=z into original eqn for x/y. Rβ • A = α g(t)f ′ (t)dt Rβ q (f ′ (t))2 + (g ′ (t))2 dt • L = α q Rβ • S = α 2πy (f ′ (t))2 + (g ′ (t))2 dt 2 = − cot θ − θ + C 2 Integration q 2.1 Definite Integrals 9 − x2 Z a Z b (cotθ = ) 1. f (x)dx = − f (x)dx x a q Zba 3.1 Cycloid 9 − x2 2. f (x)dx = 0 x = r(θ − sin θ), y = r(1 − cos θ) −1 x = − − sin +C Zab x 3 3.1.1 Area 3. cdx = c(b − a) a Z 2π 2.6 Integration by Partial Fractions 4. If = f (x)], then A = r(1 − cos θ)r(1 − cos θ)dθ Z af is even [f (−x) Z a P (x) , if deg(P ) > deg(Q), must long f (x) = 0 Q(x) f (x)dx = 2 f (x)dx Z 2π 0 −a R x3 +x 2 2 divide Q into P. Ex. dx (1 − cos θ) dθ = r 5. If x−1 Z af is odd [f (−x) = −f (x)], then 0 Z 2π f (x)dx = 0 2 2 x2 + x + 2 −a (1 − 2 cos θ + cos θ)dθ = r 0 x−1 x3 +x Z 2π 1 2 − x3 + x2 = r [1 − 2 cos θ + (1 + cos 2θ)]dθ 0 2 x2 + x r = 3y/r r x 2 +y 2 2 = 3y − 3y = 0 r = tan θ sec θ = sin θ 1 cos θ cos θ yr r yr y/r 1 = = x/r x/r xr x x2 y 2 , y = x 1 = x2 r = 5 Sequences • convergent: limit exists, divergent: limit DNE • if lim n → ∞|an | = 0 then lim n → ∞an = 0