Download 1 Definitions, Identities, Basic Ops 1.1 Exponents and Radicals 1.1.1

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