u v u v u m ( ) - 1.cdn.edl.io cot cot 2 tan sin 2 ... a b a na b + + + ⋅ ... any cylinder or...
TRANSCRIPT
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Definition of the Six Trigonometric Functions
Right triangle definitions, where 0 < θ < π / 2.
.
.tan
.
.cos
.
.sin
adj
opp
hyp
adj
hyp
opp
=
=
=
θ
θ
θ
.
.cot
.
.sec
.
.csc
opp
adj
adj
hyp
opp
hyp
=
=
=
θ
θ
θ
Circular Function Definitions, where θ is any angle
x
y
r
x
r
y
=
=
=
θ
θ
θ
tan
cos
sin
y
x
x
r
y
r
=
=
=
θ
θ
θ
cot
sec
csc
Units Circle
707.02
1
2
2== 866.0
2
3= 500.0
2
1=
Common Reference Triangles
Reciprocal Identities
xx
xx
tan
1cot
cot
1tan
=
=
xx
xx
sec
1cos
csc
1sin
=
=
xx
xx
cos
1sec
sin
1csc
=
=
Tangent and Cotangent Identities
x
xx
cos
sintan =
x
xx
sin
coscot =
Hyperbolic Functions
2sinh
xxee
x−−
= 2
coshxx
eex
−+=
x
xx
cosh
sinhtanh =
xxeex
hx−−
==2
sinh
1csc
xxeex
hx−+
==2
cosh
1sec
x
x
xx
sinh
cosh
tanh
1coth ==
Sum and Difference Formulas
( )( ) vuvuvu
vuvuvu
sinsincoscoscos
sincoscossinsin
mm =
±=±
( )vu
vuvu
tantan1
tantantan
m
±=±
Double-Angle Formulas
uuu cossin22sin = u
uu
2tan1
tan22tan
−=
uuuuu2222 sin211cos2sincos2cos −=−=−=
Power Reducing Formulas
( )uu
u 2cos12
1
2
cos21sin 2 −=
−=
u
uu
2cos1
2cos1tan 2
+
−=
( )uu
u 2cos12
1
2
cos21cos2 +=
+=
x
x
x
xx
2cos1
2sin
2sin
2cos1tan
+=
−=
Sum-to-Product Formulas
−
+−=−
−
+=+
−
+=−
−
+=+
2sin
2sin2coscos
2cos
2cos2coscos
2sin
2cos2sinsin
2cos
2sin2sinsin
vuvuvu
vuvuvu
vuvuvu
vuvuvu
Reduction Formulas
( )( ) xx
xx
csccsc
sinsin
−=−
−=−
( )( ) xx
xx
tantan
coscos
−=−
−=−
( )( ) xx
xx
cotcot
secsec
−=−
−=−
Pythagorean Identities
1cossin 22 =+ xx xx22 sectan1 =+
xx22 csccot1 =+
wwwwww.Equationsheets.comww.Equationsheets.comww.Equationsheets.comww.Equationsheets.com Fractions
bd
ac
d
c
b
a
bd
bcad
d
c
b
a
=⋅
+=+
bc
ad
c
d
b
a
d
cb
a
=⋅= b
a
b
a
b
a
−=−=
−
( ) ( )( ) ( )
( ) ( )( ) ( )
( )( )bdfgeh
fhbcad
bdfh
bdfh
hgfe
dcba
hgfe
dcba
+
+=⋅
+
+=
+
+
//
//
//
//
Exponents
m
m
nm
n
m
mnmn
AA
AA
A
AAA
1=
=
=
−
−
+
( )( )
n
nn
nnn
nmmn
B
A
B
A
BAAB
AA
=
=
=
( )
nn
mnn mnm
AA
AA
AAA
=
=
==
1
2/1
/
( )010 ≠= AA n
n
n
B
A
B
A= nnn BAAB =
Logarithms
log = common logarithm (logarithm to the base 10) = 10log
ln = natural logarithm (logarithm to the base e) = elog
e = 2.71828182846
Ae
BA
A
A
B
=
=
=
ln
log
10
A
BA
Ae
A
B
=
=
=
10log
ln
01ln1log
10 log
ln
==
=
=
A
Ae
A
A
xBA
e
=
=
=
1ln
110log
BAB
A
BAAB
logloglog
logloglog
−=
+=
AnA
AA
n loglog
log1
log
=
−=
B
Ax
log
log=
B
UU
A
A
Blog
loglog = xx 10loglog = xx elogln =
( )( ) AAA log30258509.2log10lnln ==
( )( ) ( ) AAAeA ln43429448.0lnlnlnloglog ===
Law of Sines Law of Cosines
C
c
B
b
A
a
sinsinsin== Cabbac cos2222 −+=
CabBacAbcArea sin2
1sin
2
1sin
2
1===
°=++ 180CBA
Similar Triangles
c
c
b
b
a
a '''==
Product to Sum Formulas
( ) ( )[ ]
( ) ( )[ ]
( ) ( )[ ]
( ) ( )[ ]vuvuvu
vuvuvu
vuvuvu
vuvuvu
−−+=
−++=
++−=
+−−=
sinsin2
1sincos
sinsin2
1cossin
coscos2
1coscos
coscos2
1sinsin
Cofunction Identities
xx
xx
xx
csc2
sec
sec2
csc
cos2
sin
=
−
=
−
=
−
π
π
π
xx
xx
xx
tan2
cot
cot2
tan
sin2
cos
=
−
=
−
=
−
π
π
π
xx
xx
xx
cos2
sin
sin2
cos
cos2
sin
=
+
=
−
−=
−
π
π
π
2
cos1
2sin
2
cos1
2cos
sin2
cos
xx
xx
xx
−±=
+±=
−=
+
π
x
xx
sin
cos1
2tan
−=
Quadradic Equation and Quadradic Formula
)0(
02
≠
=++
a
cbxax
a
acbbx
2
42 −±−=
Temperature Conversion Formulas
( )
( )
( ) 67.4595
915.273
5
9
15.2739
515.27332
9
5
67.45967.4595
932
5
9
15.27315.2739
532
9
5
15.273067.4590
+°=°=+°=°
+°=°=+−°=°
−°=−°=+°=°
−°=−°=−°=°
°=°°=°
FKCR
CRFK
RKCF
KRFC
KCRF
Distance Between Points ( )11 , yxP and ( )22 , yxQ is,
( ) ( ) ( )2
21
2
21, yyxxQPd ++−=
Slope of Line Between Points 12
12
xx
yym
−
−=
Distance Between any Point ( )11 , yxP and any
line 0=++ CBxAx is,
22
11
BA
CByAxd
+
++=
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( )( )
( ) 222
22
2 yxyxyx
yxyxyx
+±=±
−+=−
( )( )( ) 32233
2233
33 yxyyxxyx
yxyxyxyx
±+±=±
+±=± m
The Greek Alphabet
Alpha
Beta
Gamma
Delta
Epsilon
Zeta
Eta
Theta θ
η
ζ
ε
δ
γ
β
α
Θ
Η
Ζ
Ε
∆
Γ
Β
Α
Iota
Kappa
Lambda
Mu
Nu
Xi
Omicron
Pi π
ο
ξ
ν
µ
λ
κ
ι
Π
Ο
Ξ
Ν
Μ
Λ
Κ
Ι
Rho
Sigma
Tau
Upsilon
Phi
Chi
Psi
Omega ϖ
ψ
χ
φ
υ
τ
σ
ρ
Ω
Ψ
Χ
Φ
Υ
Τ
Σ
Ρ
Binomial Theorem
For any positive integer n,
( ) ( )
( )( ) nnn
nnnn
bnabbannn
bann
bnaaba
+++⋅⋅
−−+
⋅
−++=+
−−
−−
133
221
...321
21
21
1
For instance,
( )
( )
( )
( ) 4322344
32233
222
1
464
33
2
babbabaaba
babbaaba
bababa
baba
++++=+
+++=+
++=+
+=+
Difference of Like Integers Powers, n > 1
( )( )122321 ... −−−−− +++++−=− nnnnnnn babbabaababa
For instance,
( )( )( )( )( )( )322344
2233
22
babbaababa
babababa
bababa
+++−=−
++−=−
+−=−
Series Formulas
...!7!5!3
sin753
+−+−=xxx
xx ...!6!4!2
1cos642
+−+−=xxx
x
...!7!5!3
sinh753
++++=xxx
xx ...!6!4!2
1cosh642
++++=xxx
x
...!4!3!2
1432
+++++=xxx
xex
( )11...,753
tan753
1 ≤≤−+−+−=−x
xxxxx
( )11...,11
1 432 <<−+++++=−
xxxxxx
( ) ( )11...,432
1ln432
≤<−+−+−=+ xxxx
xx
Simple Interest Compound Interest
tI Pr=
nt
n
rPA
+= 1
I = Interest, P = Principle, r = Rate, t = Time of Investment
A = Money Accrued, n = Number of Periods in Time
Degree Conversions
π = 3.14159265359
π2 radians = 360 degrees 2
π radians = 90 degrees
1 radian = π
180 degrees = 57.2957795131º
1 degree = 180
π radians = 0.01745329252º
To get radians from degrees, multiply the degrees by 180
π
To get degrees from radians, multiply the radians by π
180
Interpolation Equation
(b is unknown)
fed
cba?
ef
bc
de
ab
−
−=
−
−
( ) ( )( )df
decefab
−
−+−=?
Example:
21149
6?2
1421
?6
914
2?
−
−=
−
−
( ) ( )
( )667.3
921
914614212? =
−
−+−=
GPS Coordinates 1 Degree = 1° = 60’ = 3600” = 60 miles = 316800 feet
1 Minute = 1’ = 60” = 1 mile = 5280 feet
1 Second = 1” = 88 feet
Convert Degrees Minutes Seconds to Decimal Degrees
DDD° MM’ SS.sssssss” DDD.ddddddd°
(i.e. 109° 40’ 04.04832” 109.6677912°)
60
60
.
.
sssssssSSMM
DDDdddddddDDD
++=°
Convert Degrees Minutes Seconds to Degrees Decimal
Minutes
DDD° MM’ SS.sssssss” DDD° MM.mmmmmmm’
(i.e. 109.6677912° 109° 40.0647200’)
++=°
60
.'.
sssssssSSMMDDDmmmmmmmMMDDD
Convert Decimal Degrees to Degrees Minutes Decimal
Seconds
DDD.ddddddd° DDD° MM’ SS.sssssss”
( )( )
( )( ) 6060.".
.060.'
60.'.
.
⋅−⋅−=
−⋅−=
⋅−=
−=°
MMDDDdddddddDDDsssssssSS
mmmmmmmDDDdddddddDDDMM
DDDdddddddDDDmmmmmmmMM
DDDdddddddDDDDDD
Convert Decimal Degrees to Degrees Decimal Minutes
DDD.ddddddd° DDD° MMM.mmmmmmm’
( )DDDdddddddDDDmmmmmmmMM
DDDdddddddDDDDDD
−=
−=°
.'.
.
wwwwww.Equationsheets.comww.Equationsheets.comww.Equationsheets.comww.Equationsheets.com Triangle
bhArea
ah
2
1
sin
=
= θ
Law of Cosines
θcos2222abbac −+=
Right Triangle
Pythagorean Theorem 222
bac +=
Equilateral Triangle
24
3
2
3
2 shsArea
sh
==
=
3
2hs =
Parallelogram
bhArea =
Trapezoid
( )bah
Area +=2
Circle
rDnceCircumfere
DDrArea
DDr
rD
ππ
πππ
2
42
22
1
2
22
2
==
=
==
==
=
Sphere
2
3
4
3
4
rAreaSurface
rVolume
π
π
=
=
Circular Ring
( p = average radius, w
= width of ring)
( )22 rRArea −= π
pwπ2=
Sector of a Circle
(θ in Radians)
2
2
2
2
2
==
D
rArea
θθ
28
2rsD
==θ
2
Drs
θθ ==
Sector of a Circular Ring
( p = average
radius, w = width of
ring,θ in Radians)
pwArea θ=
Ellipse
abcVolumeEllipsoid
banceCircumfere
abArea
π
π
π
4
3
22
22
=
+≈
=
Right Circular Cone
3
2hr
Volumeπ
=
Lateral Surface Area =
rshrr ππ =+ 22
Any Cone or Pyramid
(A = Area of Base)
3
AhVolume =
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Right Circular Cylinder
hrVolume2π=
Lateral Surface Area = rhπ2
Wedge
(A = area of upper
face,
B = area of base)
θsecBA =
Frustum of Right Circular Cone
( )3
22hRrRr
Volume++
=π
Lateral Surface Area =
( )rRs +π
Any Cylinder or Prism with Parallel Bases
bhV =
b = area
of base
Prefixes for Powers of Ten
Power
24
21
18
15
12
9
6
3
2
1
0
1
2
3
6
9
12
15
18
21
24
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
−
−
−
−
−
−
−
−
−
−
Scientific
1E-24
1E-21
1E-18
1E-15
1E-12
1E-9
1E-6
1E-3
1E-2
1E-1
1E0
1E1
1E2
1E3
1E6
1E9
1E12
1E15
1E18
1E21
1E24
Actual Number
0.000,000,000,000,000,000,000,000,1
0.000,000,000,000,000,000,000,1
0.000,000,000,000,000,001
0.000,000,000,000,001
0.000,000,000,001
0.000,000,001
0.000,001
0.001
0.01
0.1
1
10
100
1,000
1,000,000
1,000,000,000
1,000,000,000,000
1,000,000,000,000,000
1,000,000,000,000,000,000
1,000,000,000,000,000,000,000
1,000,000,000,000,000,000,000,000
Abbrev.
y
z
a
f
p
n
µ
m
c
d
---
D
h or H
k or K
M
G
T
P
E
Z
Y
Name
yocto
zepto
atto
femto
pico
nano
micro
milli
centi
deci
---
deka
hecto
kilo
mega
giga
tera
peta
exa
zeta
yotta
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Some Fundamental Constants **Numbers in parenthesis represent how many total decimal places it is carried out to be nearly exact.
Atomic Mass Unit u 1.660 540 2(10) x 2710− kg
931.494 32(28) MeV/ 2c
Avogadro’s Number AN 6.022 136 7(36) x 2310 1)( −mol
Average Earth-Moon Distance 3.84 x 810 m
1.259 842 519 690 x 910 ft
Average Earth-Sun Distance 1.496 x 1110 m
4.908 136 482 940 x 1110 ft
Average Radius of the Earth 6.37 x 610 m
2.089 895 013 120 x 710 ft
Bohr Magneton e
Bm
e
2
h=µ 9.274 015 4(31) x 2410− J/T
Bohr Radius ee kem
a2
2
0
h= 0.529 177 249(24) x 1010− m
Boltzmann’s Constant A
BN
Rk = 1.380 658(12) x 2310− J/K
7.270 063 x 2710− Btu/Rº
Compton Wavelength cm
h
e
C =λ 2.426 310 58(22) x 1210− m
Density of Air (20ºC and 1 atm) 1.20 kg/ 3m
0.074 913 600 lbm/ 3ft
Density of Water (20ºC and 1 atm) 1.00 x 310 kg/ 3m
62.428 lbm/ 3ft
Deuteron Mass dm 3.343 586 0(20) x 2710− kg
2.013 533 214(24) u
Dirac’s Constant π2
h=h 1.054 572 66(63) x 3410− J·s
2.77 505 62 x 4110− Btu·hr
Electron Mass em 9.109 389 7(54) x 3110− kg
5.485 799 03(13) x 410− u
0.510 999 06(15) MeV/ 2c
Electron-Volt eV 1.602 177 33(49) x 1910− J
Elementary Charge e 1.602 177 33(49) x 1910− C
Faraday’s Constant F 964785.309 C/gmol
Gas Constant R 8.314 510(70) J/K·mol
10.731 640 psia/ 3ft
Gravitational Constant G 6.672 59(85) x 1110− N· 2m / 2kg
3.322 80 x 1110− lbf· 22 / lbft
Gravity g 9.806 650 m/ 2s
(Free-Fall Acceleration) 32.174 ft/ 2s
Hydrogen Ground State 0
2
2
24
022 a
kekemE eee ==
h 18.605 98(40) eV
Josephson Frequency- he /2 4.835 976 7(14) x 1410 Hz/V
voltage Ratio
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Magnetic Flux Quantum e
h
20 =Φ 2.067 834 61(61) x 1510− Wb
Mass of the Earth 5.98 x 2410 kg
Mass of the Moon 7.36 x 2210 kg
Mass of the Sun 1.99 x 3010 kg
Neutron Mass nm 1.674 928 6(10) x 2710− kg
1.008 664 904(14) u
939.565 63(28) MeV/ 2c
Nuclear Magneton p
nm
e
2
h=µ 5.050 786 6(17) x 2710− J/T
Permeability of Free Space 0µ 4π x 710− N/ 2A (exact)
Permittivity of Free Space 2
0
0
1
cµε = 8.854 187 817 x 2710− 2
C /N· 2m (exact)
Photon Wavelength 0λ 1239.8425 nm
12398.425 o
A
Photon Frequency 0f 2.417 988 3 x 1410 Hz
Planck’s Constant h 6.626 075(40) x 3410− J·s
1.744 529 9 x 4010− Btu·hr
Proton Mass pm 1.672 623(10) x 2710− kg
1.007 276 470(12) u
938.272 3(28) MeV/ 2c
Quantized Hall Resistance 2
e
h 25812.805 6(12) Ω
Rydberg Constant HR 1.097 373 153 4(13) x 710 1−m
Speed of Light in Vacuum c 2.997 924 58 x 810 m/s (exact)
9.835 710 56 x 810 ft/s
Standard Atmospheric Pressure 1.013 x 510 pa
14.692 322 832 psi
Stefan-Boltzmann Constant σ 5.670 510 x 810 W/ 42Km ⋅
1.712 340 x 910− Btu/h· 2ft ·º 4R
Wein’s Constant 3c 0.002 897 756 m·K
5215.961 µ · ºR
CONVERSIONS
LENGTH m = Meter cm = Centimeter mm = Millimeter yd = Yard ft = Feet (foot)
in = Inch Mpc = Megaparsec pc = Parsec lyr = Light Year au = Astronomical Unit
km = Kilometer mi = Mile (International) nmi = Nautical Mile miUS = US Statute Mile chain = Chain
rd = Rod fath = Fathom ftUS = US Survey Foot mil = Mil µ = Micron
A = Angstrom fermi = Fermi
1 m = 100 cm = 1000 mm = 1.0936133 yd = 3.28084 ft = 39.37008 in = 0.001 km = 0.00062137 mi = 0.19884 rd = 0.4568056 fath = 3.280833 ftUS
= 1000,000 µ = 10,000,000,000 A = 1,000,000,000,000,000 fermi
1 cm = 0.01 m = 10 mm = 0.328084 ft = 0.393700787 in = 0.00001 km = 0.000006214 mi = 0.0004971 chain = 0.0019884 rd = 393.700787 mil
= 10,000 µ = 100,000,000 A = 10,000,000,000,000 fermi
1 mm = 0.001 m = 0.1 cm = 0.00328084 ft = 0.03937008 in = 0.000001 km = 0.00000062137 mi = 0.00004971 chain = 0.00019884 rd = 39.3700787 mil
= 1000 µ = 1,000,000 A = 1,000,000,000,000 fermi
1 yd = 0.9144 m = 91.44 cm = 914.4 mm = 3 ft = 36 in = 0.0009144 km = 0.000568182 mi = 0.181818 rd = 0.5 fath = 2.999994 ftUS = 36,000 mil
= 914400 µ = 9,144,000,000 A = 914,400,000,000,000 fermi
1 ft = 0.3048 m = 30.48 cm = 304.3 mm = 0.3333 yd = 12 in =0.0003048 km = 0.0001893939 mi = 0.060606 fd = 0.16667 fath = 0.999998 ftUS
= 12,000 mil = 304,800 µ = 3,048,000,000 A = 304,800,000,000,000 fermi
wwwwww.Equationsheets.comww.Equationsheets.comww.Equationsheets.comww.Equationsheets.com 1 in = 0.0254 m = 2.54 cm = 25.4 mm = 0.02778 yd = 0.083333 ft = 0.0000254 km = 0.00001578383 mi = 0.0050505 rd = 0.01389 fath = 1000 mil
= 25,400 µ = 254,000,000 A = 25,400,000,000,000 fermi
1 Mpc = 1,000,000 pc = 3,261,634.07982 lyr = 206,264,806,247 au = 3.086E19 km = 1.9174E19 mi
1 pc = 0.000001 Mpc = 3.26163497 lyr = 206,264.806247 au = 3.086E13 km = 1.9174E13 mi
1 lyr = 9.46053E15 m = 1.034616E16 yd = 3.10385E16 ft = 0.306594785168 pc = 63,239.713959 au = 9.46053E12 km = 5.8785E12 mi
1 au = 149,597,900,000 m = 290,806,758,530 ft = 0.00000485 pc = 149,597,900 km = 92,955,825.4791 mi
1 km = 1000 m = 100,000 cm = 1,000,000 mm = 1093.61329834 yd = 3280.83989502 ft = 39,370.0787402 in = 3.24078E-14 pc = 1.057E-13 lyr
= 6.6846E-9 au = 0.62137119224 mi = 0.539956803458 nmi = 0.621369949498 miUS = 49.7095959596 chain = 198.838383 rd = 546.80556 fath
= 3280.83333 ftUS
1 mi = 1609.344 m = 160,934.4 cm = 1,609,344 mm = 1760 yd = 5280 ft = 63,360 in = 5.21553E-14 pc = 1.70111E-13 lyr = 1.07578E-8 au
= 1.609344 km = 0.868976241901 nmi = 0.999998 miUS = 79.99984 chain = 319.99936 rd = 879 99824 fath = 5279.98944 ftUS
1 nmi = 1852 m = 2025.37182852 yd = 6076.11548556 ft = 1.852 km = 1.15077944802 nmi = 1.15077714646 miUS = 92.0621717168 chain
= 368.248686867 rd = 1012.68388888 fath = 6076.10333328 ftUS
1 miUS = 1609.34721869 m = 1760.00352 yd = 5280.01056 ft = 1.60935 km = 1.000002 mi = 79.99999 chain = 319.99999 rd = 879.99999 fath
1 chain = 20.1168402337 m = 2011.68402337 cm = 20,116.8402337 mm = 22.000044 yd = 66.000132 ft = 792.001584 in = 0.02011684 km
= 0.0125 mi = 4 rd = 11 fath = 66 ftUS = 792,001.584003 mil
1 rd = 5.02921 m = 502.921 cm = 5029.21 mm = 5.500011 yd = 16.500033 ft = 198.000396 in = 0.00502921 km = 0.003125 mi = 0.25 chain
= 2.75 fath = 16.5 ftUS = 198000.396 mil
1 fath = 1.8288 m = 2.000004 yd = 6.000012 ft = 0.0018288 km = 0.00136366 mi = 0.090909 chain = 0.363636 rd = 6 ftUS = 72,000.144 mil
1 ftUS = 0.3048 m = 0.333334 yd = 1.000002 ft = 12.000024 in = 0.0003048 km = 0.000189394 mi = 0.0151515 chain = 0.060606 rd = 12,000.024 mil
1 mil = 0.0000254 m = 0.00254 cm = 0.0254 mm = 0.00008333 ft = 0.001 in = 24.5 µ = 254,000 A = 25,400,000,000 fermi
1 µ = 0.000001 m = 0.0001 cm = 0.001 mm = 0.00000328084 ft = 0.00003937 in = 0.03937 mil = 10,000 A = 1,000,000,000 fermi
1 A = 0.0000000001 m = 0.00000001 cm = 0.0000001 mm = 3.937E-9 in = 0.000003937 mil = 0.0001 µ = 100,000 fermi
1 fermi = 1E-15 m = 1E-13 cm = 1E-12 mm = 3.937E-14 in = 3.937E-11 mil = 0.000000001 µ = 0.00001 A
AREA m² = Meters Squared cm² = Centimeters Squared B = Barn yd² = Yard squared
ft² = Feet squared in² = Inches Squared km² = Kilometers Squared Ha = Hectar
A = Are mi² = Miles squared Acre = Acre miUS² = US Statute Miles Squared
1 m² = 10,000 cm² = 1.19599 yd² = 10.7639 ft² = 1550.0031 in² = 3.86102E-7 mi² = 2.471044E-4 Acre = 1E28 B
1 cm² = 0.155 in² = 1.0764E-3 ft²
1 yd² = 0.83613 m² = 9 ft² = 1296 in²
1 ft² = 0.11111 yd² = 144 in² = 929.0304 cm² = 0.092903 m²
1 in² = 6.4516 cm² = 6.9444E-3 ft² = 0.00064516 m²
1 km² = 100 Ha = 10,000 A = 247.1044 Acre = 0.386102 mi²
1 Ha = 0.01 km² = 100 A = 2.471044 Acre = 3.86102E-3 mi²
1 A = 2.471044E-4 Acre = 0.01 Ha = 0.0001 km² = 3.86102E-5 mi² = 1E30 B
1 mi² = 2.589988 km² = 639.99744 Acre = 0.999996 miUS² = 3,097,600 yd² = 27,878,400 ft² = 2,589,933.11034 m² = 258.998811 Ha
1 miUS² = 1.000004 mi²
1 Acre = 4046.8726 m² = 4840.01936 yd² = 43,560.17424 ft² = 2.5625E-3 mi² = 4.046873E-3 km² = 0.4046873 Ha = 40.46873 A
1 B = 1E-28 m²
VOLUME m³ = Meters Cubed st = Stere cm³ = Centimeters Cubed yd³ = Yards Cubed \ft³ = Feet Cubed
in³ = Inches Cubed L = Liter galUK = Gallon (UK) galC = Gallon (Canadian) gal = Gallon (US)
pt = Pint ml = Milliliter cu = Cup (US) oz = Fluid Ounce (US) ozUK = Fluid Ounce (UK)
tbsp = Tablespoon tsp = Teaspoon bbl = Barrel bu = Bushel pk = Peck
fbm = Board Foot qt = Quart
1 m³ = 1 st = 1,000,000 cm³ = 1.30795 yd³ = 35.314667 ft³ = 61023.7440946 in³ = 1000 L = 1,000,000 ml = 264.172 gal = 1056.68821 qt = 2113.37642 pt
= 4226.7528377 cu = 33,814.0227 oz = 6.28981 bbl = 28.3775934 bu = 113.510374 pk
1 st = 1 m³ (same as above)
1 cm³ = 0.000001 m³ = 0.000035315 ft³ = 0.061024 in³ = 1000 m³ = 0.000264172 gal = 0.0010567 qt = 0.0021134 pt = 1 ml = 0.004226753 cu
= 0.033814 oz = 0.067628 tbsp = 0.202884 tsp
1 yd³ = 0.76455486 m³ = 764,554.857984 cm³ = 27 ft³ = 46,656 in³ = 764.55486 L = 201.974026 gal = 807.8961 qt = 1615.79221 pt = 3231.584416 cu
= 25,852.67532 oz = 4.808905 bbl = 21.696227 bu = 86.784907 pk
1 ft³ = 0.02832 m³ = 28,316.8466 cm³ = 0.037037 yd³ = 1728 in³ = 28.31685 L = 7.48052 gal = 576/77 gal = 29.92208 qt = 59.844156 pt = 119.68831 cu
= 957.50649 oz = 0.1781076 bbl = 0.803564 bu = 3.214256 pk = 12 fbm
1 in³ = 16.387064 cm³ = 0.000578704 ft³ = 0.016387 L = 0.004329 gal = 0.017316 qt = 0.034632 pt = 16.387064 ml = 0.069264 cu = 0.5541125 oz
= 1.108225 tbsp = 3.32467 tsp
1 L = 0.001 m³ = 1000 cm³ = 61.023744 in³ = 0.264172 gal = 1.0566882 qt = 2.113376 pt = 1000 ml = 4.226753 cu = 33.814 oz = 67.628 tbsp
= 202.88414 tsp
1 gal UK = 1.00000044 gal C = 1.2009505 gal
1 gal C = 0.99999956 gal UK = 1.2009499 gal
1 gal = 0.83267381 gal UK = 0.8326742 gal C = 0.003785 m³ = 3,785.411784 cm³ = 0.13368055 ft³ = 231 in³ = 3.7854 L = 4 qt = 8 pt = 16 cu = 128 oz
= 256 tbsp = 768 tsp = 3,785.411784 ml = 0.0238952 bbl = 0.1074209 bu = 0.4296835 pk
1 qt = 0.25 gal = 2 pt = 4 cu = 32 oz = 64 tbsp = 192 tsp = 0.00946353 m³ = 946.352946 cm³ = 57.75 in³ = 0.946352946 L = 946.352946 cm³
= 0.02685522 bu = 0.107451 pk
1 pt = 0.125 gal = 0.5 qt = 2 cu = 16 oz = 32 tbsp = 96 tsp = 473.176473 cm³ = 28.875 in³ = 0.473176 L = 473.176473 ml = 0.0134276 bu = 0.05371 pk
1 ml = 1 cm³ (see 1 cm³ above)
1 cu = 8 oz = 16 tbsp = 48 tsp = 0.25 qt = 0.5 pt = 0.0625 gal = 236.5882365 cm³ = 236.5882365 ml = 14.4375 in³ = 0.23658824 L
wwwwww.Equationsheets.comww.Equationsheets.comww.Equationsheets.comww.Equationsheets.com 1 oz = 1.0408423 ozUK = 2 tbsp = 6 tsp = 29.57353 ml = 29.57353 cm³ = 0.029574 L = 0.0078125 gal = 0.03125 qt = 0.0625 pt = 1.8046875 in³
1 ozUK = 0.96076036 oz
1 tbsp = 3 tsp = 0.5 oz = 0.0625 cu = 14.7867648 ml = 14.7867648 cm³ = 0.0147868 L = 0.00390625 gal = 0.015625 qt = 0.03125 pt = 0.90234375 in³
1 tsp = 0.333333 tbsp = 0.1666667 oz = 0.020833334 cu = 4.9289216 ml = 4.9289216 cm³ = 0.00492892 L = 0.00130208 gal = 0.0104167 pt = 0.3008 in³
1 bbl = 4.51168 bu = 18.0467072 pk = 67.375 fbm = 0.1589873 m³ = 5.61458334 ft³ = 0.20794753 yd³ = 42 gal = 158.987295 L = 168 qt = 336 pt
= 672 cu = 5376 oz = 10,752 tbsp = 32,256 tsp
1 bu = 0.221647 bbl = 4 pk = 14.93347 fbm = 0.03523907 m³ = 1.244456 ft³ = 0.046091 yd³ = 9.309177 gal = 35.23907 L = 37.23671 qt = 74.4734196 pt
1 pk = 0.25 bu = 0.05541177 bbl = 3.733368 fbm = 0.0088977 m³ = 0.311114 ft³ = 0.0115227 yd³ = 2.3272944 gal = 8.8097675 L = 9.3091774 qt
1 fbm = 0.2678546 pk = 0.066964 bu = 0.0148423 bbl = 0.00235074 m³ = 0.083334 ft³ = 0.00308642 yd³ = 0.6233766 gal = 2.35974 L = 2.4935065 qt
TIME yr = Year day = Day hr = Hour min = Minute s = Second
1 yr = 365.242198781 day = 8765.81277074 hr = 525,948.766244 min = 31,556,925.9746 s
1 yr = 365 day = 8760 hr = 525,600 min = 31,536,000 s
1 day = 24 hr = 1400 min = 86,400 s
1 hr = 60 min = 3600 s
VELOCITY ( SPEED ) m/s = Meters per Second ft/s = Feet per Second kph = Kilometers per Hour cm/s = Centimeters per Second
mph = Miles per Hour c = Speed of Light mi/m = Miles per Minute knot = Nautical Miles per Hour
1 m/s = 100 cm/s = 3.28084 ft/s = 3.6 kph =2.23693629 mph = 1.9438445 knot
1 cm/s = 0.01 m/s = 3.28084E-2 ft/s = 0.036 kph = 2.23693629E-2 mph = 1.9438445E-2 knot
1 ft/s = 0.3048 m/s = 30.48 m/s = 1.09728 kph = .68181818 mph = .5924838 knot
1 kph = .277778 m/s = 27.77778 cm/s = .9113444 ft/s = 0.62137119 mph = 0.5399568 knot
1 mph = 0.44704 m/s = 44.704 cm/s = 1.466667 ft/s = 22/15 ft/s = 1.609344 kph = 0.86897624 knot
1 knot = 0.51444 m/s = 51.4444 cm/s = 1.68781 ft/s = 1.852 kph = 1.14077945 mph
1 c = 299,792,458 m/s = 983,571,056.43 ft/s = 1,079,252,848.8 kph = 670,616,629.384 mph = 582,749,918.358 knot
1 mi/m = 26.8224 m/s = 60 mph = 88 ft/s = 52.1386 knot
ACCELERATION m/s² = Meters per Second Squared ft/s² = Feet per Second Squared cm/s² = Centimeters per Second Squared
1 m/s² = 3.280835 ft/s² = 100 cm/s²
1 ft/s² = 0.3048 m/s² = 30.48 cm/s²
1 cm/s² = 0.01 m/s²
9.80665 m/s² = 32.174 ft/s²
MASS kg = Kilogram g = Gram lb = lbm = Pound (mass) oz = Ounce slug = Slug
lbt = Troy Pound ton = Short Ton tonUK = Long Ton UK T = Metric Ton ozt = Troy Ounce
ct = Carat grain = Grain U = Unified Atomic Mass
1kg = 1000 g = 2.20462 lb = 35.2739 oz = 0.06852 slug = 0.001 T = 5000 ct = 6.02E26 U
1 g = 0.001 kg = 0.03527 oz = 0.000001 T = 5 ct = 15.4324 grain
1 lb = 0.45359 kg = 435.592 g = 16 oz = 0.03108 slug = 1.21528 lbt = 0.0005 ton = 0.0004536 T = 14.5833 ozt = 7000 grain = 2.73E26 U
1 oz = 0.02835 kg = 28.3495 g = 0.91145 ozt = 141.747 ct = 437.50 grain
1 slug = 1
ft
slb2⋅ = 14.5939 kg = 32.174 lb = 514.785 oz = 469.205 ozt
1 lbt = 373.242 g = 0.82286 lb = 13.1657 oz = 12 ozt = 5760 grain
1 ton = 907.185 kg = 2000 lb = 32000 oz = 14,000,000 grain
1 tonUK = 1016.05 kg = 2240 lb = 35,840 oz = 1.12 ton = 1.01605 T = 6.02E29 U
1 T = 1000 kg = 2204.62 lb = 1.10231 ton = 5,000,000 ct
1 ozt = 31.1035 g = 1.09714 oz = 155.517 ct = 480 grain
1 ct = 0.2 g = 0.0004409 lb = 0.00705 oz = 0.0000002 T = 3.06865 grain
1 grain = 0.06479 g = 0.02286 oz = 0.32399 ct
1 U = 1.6E-27 kg = 3.66E-27 lb = 1.6E-30 T = 931.5 2c
MeV
FORCE N = Newton dyn = Dyne lbf = Pound (force) gf = Gram (force) kip = kiloPound (force)
pdl = Poundal
1 N = 12
s
mkg ⋅ = 1E5 dyn = 0.2248 lbf = 101.97162 gf = 7.233 pdl
1 dyn = 1E-5 N = 2.2481E-6 lbf
1 lbf = 4.448222 N = 444822.161526 dyn = 9.001 kip
1 gf = 0.00980665 N
1 kip = 1000 lbf
wwwwww.Equationsheets.comww.Equationsheets.comww.Equationsheets.comww.Equationsheets.com 1 pdl = 0.138254954 N
ENERGY, WORK, HEAT J = Joule erg = Erg Kcal = Cal = kilo-Calorie cal = Calorie Btu = British Thermal Unit
ft·lb = Foot – Pound therm = Therm (EEC) MeV = Mega electron Volt eV = electron Volt kWh = kiloWatt hour
1 J = 1 N·m = 1 W·s = 1 2
2
s
mkg ⋅ = 10,000,000 erg = 0.0002388 Kcal = 0.238846 cal = 0.00094782 Btu = 0.737562 ft·lb = 9.47813E-9 therm
= 6.2415E18 eV = 2. 777777778E-7 kWh
1 erg = 0.0000001 J = 624150.636307 MeV = 624,150,636,307 eV
1 Kcal = 4186.8 J = 41,868,000,000 erg = 1000 cal = 3.968321 Btu = 3088.0252066 ft·lb = 0.00003968 therm
1 cal = 4.1868 J = 41,868,000 erg = 0.0.001 Kcal = 0.00396832 Btu = 3.0880252 ft·lb = 3.9683E-8 therm
1 Btu = 1055.05585 J = 0.251996 Kcal = 251.99576 cal = 778.16926 ft·lb = 0.000010002 therm = 0.000293071 kWh
1 ft·lb = 1.35582 J = 0.00032383 Kcal = 0.32383155 cal = 0.00128507 Btu = 3.766161E-7 kWh
1 therm = 105,506,000 J = 1.05506E15 erg = 25,199.67517 Kcal = 25,199,675.1696 cal = 100,000.393096 Btu = 77,817,232.1218 ft·lb = 29.300111111 kWh
1 MeV = 0.000001602177 erg = 1,000,000 eV
1 eV = 1.602177eeE-19 J = 1.60217733E-12 erg = 0.000001 MeV
1 kWh = 3,600,000 J = 3.6 MJ = 2,655,220 ft·lb
1 MMBtu = 1 dekatherm
POWER W = Watt Hp = horsepower J= Joule kW = kiloWatt btu/hr = British Thermal Units per Hour
1 W = 1 s
J = 1 s
mN ⋅ = 0.738
s
lbft ⋅ = 3.41297
hr
btu = 1.3410221E-3 hp = 10,000,000
s
erg
1 hp =745.69987 W = 550 s
lbft ⋅ = 33,000
min
lbft ⋅ = 2545.05076
hr
btu = 0.746 kW
1 kW = 1000 W = 1.3410221 hp = 3412.96928 hr
btu = 737.562
s
lbft ⋅
1 hr
btu = 0.293 W
PRESSURE pa = Pascal atm = Atmosphere bar = Bar torr = Torr psi = Pound per Square Inch
inHg = Inches of Mercury inH20 = Inches of Water mmHg = Millimeters of Mercury psf = Pound per Square Foot
1 pa = 12
m
N = 9.86923E-6 atm = 0.00001 bar = 0.000145 psi = 0.00750062 torr = 0.00750062 mmHg = 0.0002953 inHg = 0.00401865 inH20
1 atm = 101,325 pa = 1.01325 bar = 14.6959487755 psi = 760 torr = 760 mmHg = 29.92126 inHg = 407.189359 inH20
1 bar = 100,000 pa = 0.9869233 atm = 14.503774 psi = 750.061683 torr = 750.061683 mmHg = 29.5299875 inHg = 401.864652 inH20
1 psi = 6894.757293 pa = 0.068046 atm = 0.0689476 bar = 51.714933 torr = 51.714933 mmHg = 2.036021 inHg = 27.7075924 inH20 = 144 psf
1 torr = 1 mmHg (same as below)
1 mmHg = 133.32237 pa = 0.001315789 atm = 0.00133322 bar = 0.0193368 psi = 0.03937 inHg = 0.5357755 inH20
1 inHg = 3386.38816 pa = 0.033421053 atm = 0.03386388 bar = 0.491154077 psi = 25.4 torr – 25.4 mmHg = 13.608697 inH20
1 inH20 = 248.84 pa = 0.00245586 atm = 0.0024884 bar = 0.0360912 psi = 1.8664535 torr = 1.8664535 mmHg = 0.073482421 inHg
1 Mpa = 1 2
mm
N
1 kpa = 1 2
m
kN
1 2
cm
dyn = 0.1 pa
TEMPERATURE ºC = degrees Celsius ºF = degrees Fahrenheit ºK = degrees Kelvin ºR = degrees Rankine
ºC = ( ) 15.27315.2739
532
9
5−°=−°=−° KRF
ºF = 67.45967.4595
932
5
9−°=−°=+° RKC
ºK = ( ) 15.2739
515.27332
9
5+°=°=+−° CRF
ºR = ( ) 67.4595
915.273
5
9+°=°=+° FKC
RF °=° 67.4590 KC °=° 15.2730
wwwwww.Equationsheets.comww.Equationsheets.comww.Equationsheets.comww.Equationsheets.com ELECTRICITY V = Volt A = Ampere C = Coulomb Ω = Ohm F = Farad
W = Watt Fdy = Faraday H = Henry mho = Mho S = Siemen
T = Tesla Wb = Weber
1 V = 1 2
2
sA
mkg
⋅
⋅ = 1 Ω⋅A
1 A = 1 Ω
V
1 C = 0.0000103641 Fdy = 1 sA ⋅
1 Ω = 1 32
2
sA
mkg
⋅
⋅ = 1 A
V
1 F = 1 2
42
mkg
sA
⋅
⋅
1 W = 1 3
2
s
mkg ⋅ = 1 s
J = 1 s
mN ⋅
1 Fdy = 96,487 C = 96,487 sA ⋅
1 H = 1 22
2
sA
mkg
⋅
⋅
1 mho = 1 S = 1 2
32
mkg
sA
⋅
⋅
1 S = 1 mho = 1 2
32
mkg
sA
⋅
⋅
1 T = 1 2sA
kg
⋅
1 Wb = 1 2
2
sA
mkg
⋅
⋅
ANGLES ° = Degree rad = Radian grad = Grade arcmin = Minute of Arc arcs = Second of Arc
1° = 0.0174533 rad = 1.11111111 grad = 60 arcmin = 3600 arcs
1 rad = 57.29578 ° = 63.661977 grad = 3437.74677078 acrmin = 206,264.806247 arcs
1 arcmin = 1/60 ° = 1/3437.74677078 rad = 1/54 grad = 60 arcs
1 arcs = 1/3600 ° = 0.00000484814 rad = 1/3240 grad
1 sr = 1 22 −⋅ mm
LIGHT fc = Foot Candle flam = Foot Lambert lx = Lux ph = Phot sb = Stilb
lm = Lumen cd = Candela lam = Lambert
1 fc = 10.763891 lx = 0.0010764 ph = 10.7639104167 2
m
srcd ⋅
1 flam = 0.000342626 sb = 0.0010764 lam = 3.4262591 2
m
cd
1 lx = 0.9290304 fc = 0.0001 ph = 12
m
srcd ⋅
1 ph = 929.0304 fc = 10,000 lx = 1000 2
m
srcd ⋅
1 sb = 2918.6350796 flam = 3.1415926 lam = 10,000 2
m
cd
1 lm = 1 cd·sr
1 cd = 1 candle power
1 lam = 929.0304 flam = 0.3183099 sb = 3183.09886184 2
m
cd
RADIATION Gy = Gray rad = Rad rem = Rem Sv = Sievert Bq = Becquerel
Ci = Curie R = Roentgen
1 Gy = 100 rad = 100 rem = 1 Sv = 1 2
2
s
m
1 rad = 0.01 Gy = 1 rem = 0.01 Sv
wwwwww.Equationsheets.comww.Equationsheets.comww.Equationsheets.comww.Equationsheets.com 1 rem = 0.01 Gy = 1 rad = 0.01 Sv
1 Sv = 1 Gy = 100 rad = 100 rem = 1 2
2
s
m
1 Bq = 2.7027E-11 Ci = 1 s
1
1 Ci = 37,000,000,000 Bq
1 R = 0.000258
kg
sA ⋅
VISCOSITY P = Poise St = Stokes
1 P = 1 sm
kg
⋅
1 St = 0.0001 s
m2
FREQUENCY hz = Hertz cpm = Cycles per Minute cps = Cycles per Second rev = Revolution rpm = Revolution per Minute
1 hz = 1 cps = 60 rpm
1 rpm = 1 cpm = 60
1hz
1 rev = 360 deg
1000 rpm = 60
1000 hz
1 min
rev = 60
2π s
rad
Derivatives
Remember: dudx
duu ==' , a = constant
( ) aaxdx
d=
[ ] 'auaudx
d=
[ ] 0=adx
d
[ ] 1=xdx
d
dx
du
du
dv
dx
dv=
dudxdx
du
/
1=
[ ] '' vuuvuvdx
d+= (Chain Rule)
[ ] '' vuvudx
d±=±
2
''
v
uvvu
v
u
dx
d −=
[ ] ( ) 'cossin uuudx
d=
[ ] ( ) 'sincos uuudx
d−=
[ ] ( ) 'sectan 2uuu
dx
d=
[ ] ( ) 'tansecsec uuuudx
d=
[ ] ( ) 'cotcsccsc uuuudx
d−=
[ ] ( ) 'csccot 2uuu
dx
d−=
[ ] ( )111
'sin
2
1 <<−−
=−u
u
uu
dx
d
[ ] ( )111
'cos
2
1 <<−−
−=−
uu
uu
dx
d
[ ]2
1
1
'tan
u
uu
dx
d
+=−
[ ] ( )11
'sec
2
1 >−
=−u
uu
uu
dx
d
[ ] ( )11
'csc
2
1 >−
−=−
uuu
uu
dx
d
[ ]2
1
1
'cot
u
uu
dx
d
+
−=−
( ) ( ) 'coshsinh uuudx
d=
( ) ( ) 'sinhcosh uuudx
d=
( ) 'log
log uu
eu
dx
d
=
[ ]u
uu
dx
d 'ln =
( ) ( ) 'ln uaaadx
d uu =
wwwwww.Equationsheets.comww.Equationsheets.comww.Equationsheets.comww.Equationsheets.com Integrals Remember: x is a variable, u is a variable,
v is a variable, a = constant, b = constant,
C = constant, k = constant, f(x) = function
n = integer, fraction, and/or number
m = integer, fraction, and/or number
Caxadx +=∫
∫ += Cxdx
( ) ( )∫ ∫ += Cxfkdxxkf
( )11
1
−≠++
=∫+
nCn
xdxx
nn
( )11
1
≠+−
=∫−
nCn
x
x
dxn
n
∫ += Cxx
dxln
( )∫ +−= Cxxxdxx lnln
( )∫ +−= Cxaxxdxax lnln
( ) ( ) Cdxxfdxxfb
a
a
b+−=∫ ∫
( ) ( ) ( )∫ ∫∫ +=b
a
c
b
c
adxxfdxxfdxxf
Cedxe xx +=∫
Ca
edxe
axax +=∫
( )∫ ∫∫ +=+ dvdudvdu
∫ ∫−= vduuvudv (Integration by Parts)
Ca
adxa
uu +=∫ ln
( ) ( )10ln
1≠>+=∫ aaC
a
a
bdxa
bxbx
Caxaxx
dx+=∫ lnln
ln
( )∫ ++=+
Cbxabbxa
dxln
1
( )∫ +±+=±
Caxxax
dx 22
22ln
∫ +−=−
Cxaxa
xdx 22
22
∫ +±=±
Caxax
xdx 22
22
Ca
x
aaxx
dx+=
−
−
∫1
22sec
1
Cxxx
dx+=
−∫
−1
2sec
1
( ) ( )∫ +
+=
+
Cm
axdxaxx
mmn
1
lnln
1
( )1≠m
( ) Caxa
edxxe
axax +−=∫ 1
2
∫ ∫ +−= −Cdxex
a
nex
adxex
axnaxnaxn 11
( )∫ +−= Caxa
edxxe
axax 1
2
∫ −=ba
bxdxbx
axnaxn
ln
( ) ( )10ln
1 ≠>+∫−
bbCbxdxxba
n n
∫ +=+
−Cx
x
dx 1
2tan
1
Cx
a
axa
dx+=
+∫−1
22tan
1
Cxa
xa
axa
dx+
−
+=
−∫ ln2
122
( )022 >− xa
Cax
ax
aax
dx+
+
−=
−∫ ln2
122
( )022 >− ax
( ) Cxaaxa
xdx+±±=
±∫22
22ln
2
1
( )C
x
ax
aaxx
dx+
+−=
+∫ ln1
( ) ( )( )
( )11
1
−≠++
+=+
+
∫ nCnb
bxadxbxa
nn
( ) ( )∫ ++
−=+
Cbxabbxa
dx 12
( ) ( )∫ ++
==+
Cbxabbxa
dx23
2
1
∫ =+ 222 xba
dx
Ca
xb
a
bx
b+
++
2
22
1ln1
∫ =− dxxba 222
Ca
bx
b
axba
x+++ −1
2222 sin
22
∫ +=−
−C
a
bx
bxba
dx 1
222sin
1
( ) Cbeaaca
x
bea
dx cx
cx++−=
+∫ ln1
( )∫ +−= Cxdux cossin
( )∫ += Cxdux sincos
( )∫ +−= Cxdxx coslntan
( )∫ ++−= Cxxdxx cotcsclncsc
( ) Cxxdxx ++=∫ tanseclnsec
( )∫ += Cxdxx sinlncot
( )∫ +−= Caxa
dxax cos1
sin
( )∫ += Ca
axdxax
sincos
( ) ( ) =+−=∫ Caxa
dxax cosln1
tan
( ) Caxa
+secln1
( ) ( ) =+−=∫ Caxaxa
dxax cotcscln1
csc
Cax
a+
2tan
1
( ) ( ) =++=∫ Caxaxa
dxax tansecln1
sec
Cax
a+
+
42tan
1 π
( ) ( ) Caxa
dxax +=∫ sinln1
cot
( ) Cxx
dxx +−=∫ 4
2sin
2sin 2
( ) Cxx
dxx ++=∫ 4
2sin
2cos2
( )∫ +−= Cxxdxx tantan 2
( )∫ += Cxdxx tansec2
( ) Cxdxx +−=∫ cotcsc2
( )∫ +−= Cxdxx sinlncot 2
( ) ( )C
bxabbxa
dx+
+−=
+∫ 34
3
1
( )[ ]∫ ++−=+
Cbxaabxbbxa
xdxln
12
( )( )∫ +
++
+=
+Cbxa
bxa
a
bbxa
xdxln
122
( ) ( )( )( )∫ ++−
−=+
−C
bxabnbxa
dxnn 1
1
1,
( )1≠n
( )∫ ++=+ Cbxab
dxbxa3
3
2
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( )Cx
xx
xd+=
−
− −
∫1
2csc
1
∫−=
− a
bx
abxba
dx 1
222tan
1 (x in rads)
( )0,0 >> ba
=+=−∫
−C
a
bx
abxba
dx 1
222tanh
1
Cbxa
bxa
ab+
−
+ln
2
1 (x in rads)
( ) ( )∫+
+−=
+223
2
2
bxab
bxa
bxa
xdx
( ) ( )∫+
+−=
+324
6
3
bxab
bxa
bxa
xdx
⋅=+∫ 3
2
2
1
bbxa
dxx
( )( ) ( )[ ] Cbxaabxabxa ++++−+ ln23 2
( )∫ =+
2
2
bxa
dxx
( ) ( ) Cbxaa
bxa
bxabx
b+
+−
+
+ln2
213
( )∫ =+
3
2
bxa
dxx
( )( )
( ) Cbxabxa
bxaa
b+
++
+
+ln
2
43123
( ) ( )∫ ++
++−=
+C
bxab
xbabxa
bxa
dxx33
222
4
2
3
33
∫ ++=+ 222222
2xba
xdxxba
Ca
xb
a
bx
b
a+
++
2
222
1ln2
∫ =+ bxax
dx2
Cbxax
dx
a
b
ax
bxa+
++
+− ∫2
( )∫ =+
222 xa
dx
( ) Ca
x
axaa
x++
+−1
3222tan
2
1
2
( )∫ =−
222 xa
dx
( ) Cxa
dx
axaa
x+
−+
− ∫ 222222 2
1
2
( )∫ ++
=+
Caxa
x
ax
dx
2222/322
( )∫ ++
−=+
Caxax
xdx
222/322
1
( )∫ ±±=±322222
4ax
xdxaxx
( ) Caxxa
axxa
+±+−± 224
222
ln88
( )∫ +−−=−322222
4xa
xdxxax
Ca
xaxax
a+
+− −122
2
sin8
(a >0)
=+∫ dxbxax
( ) ( )
Cb
bxabxa+
+−−2
3
15
322
=+∫ dxbxax 2
( ) ( )C
b
bxaxbabxa+
++−3
3222
105
151282
( )∫ ++=+
Cabxbbxa
xdx 2
2ln
2
1
?tan 1
2
2
+−=+∫
− Cabx
abb
a
b
x
bxa
dxx
∫ =++ 2cxbxa
dx
Cc
bcxcxbxa
c+
++++
2ln
1 2
( )04
2sin
1
2
1 >+
−
−−
−= −
cCacb
bcx
c
( )Cbxa
b
a
b
x
bxax
dx++−=
+∫ ln2
( ) ( )−
+
−=−∫ 1
2222
n
axxdxax
n
( ) ( )11
222
2
−≠+−+ ∫
−
nCdxaxn
na n
( )∫ =−
n
ax
dx
22
( )( ) ( )∫ +
−−
−−
C
ax
dx
an
axxn 2
222
22
2
( )2≠n
( ) ( )2
222
22 ++
−=−∫
+
Cn
axdxaxx
nn
∫ =− dxxa 22
Ca
xaxax +
+− −1222 sin
2
1
( )∫ +−−=− Cxadxxax2/32222
3
1
∫ =± dxax 22
( )[ ]Caxxaaxx +±+±± 22222 ln2
1
( ) Caxdxaxx +±=±∫2/32222
3
1
∫ =+=−
−C
a
x
xa
dx 1
22sin
( )0cos 221 >−+− −xaC
a
x
=+=+
−
∫ Cx
a
xa
dx 1
22sin
( ) Cxax +++ 22ln
∫ =+=−
−C
a
x
ax
dx 1
22cosh
Caxx +−+ 22ln
−−=− −
∫ a
xdx
x
xa 1
2
22
sin
Cx
xa+
− 22
( )−++=+
∫22
2
22
ln xaxdxx
xa
Cx
xa+
+ 22
Cxa
ax
axx
dx+
−=
−∫ 2
22
222
=−
∫ 222 xax
dx
Cx
xaa
a+
++−
22
ln1
∫ =− 22 xax
xdx
Cxaxa
axa +−−
−− 21 2sin
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( )∫ ++=+ 2222222 28
xaxax
dxxax
( ) Cxaxa
+++− 224
ln8
∫ =+
dxx
xa22
Cx
xaaaxa +
++−+
2222 ln
∫ =+ 22
2
xa
x
( ) Cxax
xaxa
++
+++2
ln2
2222
2
∫ ++
−=+
Cxa
xa
xax
dx2
22
222
∫ ==− −
a
xadxxax
14
222 sin8
( ) Cxaxax +−− 2222 28
1
∫ =−
dxx
xa22
Cx
xaaxa +
−+−−
2222 ln
∫ +−
−=−
Cxa
xadx
xa
x2
22
22
2
∫ =− dxax 22
Caxxa
axx
+−++− 222
22 ln22
( ) Ca
axxdxax +−=∫ 4
2sin
2sin 2
( )∫ ++= Ca
axxdxax
4
2sin
2cos2
( ) ( ) Cxaxa
dxax +−=∫ tan1
tan 2
( )∫ +−= Caxa
dxax cot1
csc2
( )∫ += Caxa
dxax tan1
sec2
( ) =+−−=∫ Cxaxa
dxax cot1
cot 2
Caxa
+− sinln1
∫ =−
dxx
ax22
Cx
aaax +−− −122 sec
=−
∫ dxx
ax2
22
Cx
axaxx +
−−−+
2222ln
∫ =−
dxax
x
22
2
Caxx
axxa
+−+−+ 22222
2ln
2
∫ =− 22 axx
dx
Cx
a
aC
a
x
a+=+ −− 11 cos
1sec
1
Ca
ax
xax
dx+
−=
−
−
∫1
2sin
2
∫ =− 22 xax
dx
Ca
axaxax
ax+
−+−
− −12
2 sin2
22
( ) ( )( )∫ +
−−=−
1
22
22
n
xaxaxdxxax
nn
( )∫ +−+
+−
Cdxxaxn
na n 22
2
21
( )( )( )
( )+
−
−−
−∫
−
2
22
2 2
2
2 an
xaxax
xax
dxa
n
( )
( ) ( )∫ +−−
−−
C
xax
dx
an
nn 2
22
22
3
( ) =∫− dxax1cos
Cxaa
axx +−−− 221 11
cos
( )∫ =− dxax1tan
( ) Cxaaxx ++−− 221 1ln2
1tan
( ) Ca
axx
a
axdxaxx +−=∫
cossinsin
2
(x in rads)
( ) Ca
axx
a
axdxaxx ++=∫
sincoscos
2
(x in rads)
( ) =+∫ dxbxaxn
( )
Cn
a
n
bxa
b
bxan
+
+−
+
+++
122
1
,
( )2,1 −−≠n
( ) ( )C
n
bxa
bdxbxa
nn
++
+=+∫
+
2
22
,
( )2−≠n
∫ =+
dxx
bxa
∫ ++
++ Cbxax
dxabxa2
tan2 1 +
−
+
−=
+∫
− Ca
bxa
abxax
dx
ln1
+++
−+=
+∫ C
abxa
abxa
abxax
dx
∫ =+
dxx
bxa2
Cbxax
dxb
x
bxa+
++
+− ∫2
+
−=− −
∫ a
axadxxaxx
13
2 sin2
2
( )( )
Cxaxaxax
+−−+
6
232 2
=−
∫ dxx
xax22
Ca
axaxax +
−+− −12 sin2
=−
∫ dxx
xax2
22
Ca
ax
x
xa+
−−
−− −1sin
22
Cx
xa
axaxx
dx+
−−=
−∫
21
2 2
( ) ( )( )∫ −
+
+−=
ba
xbadxbxax
2
coscossin
( )( )
( )22
2
cosbaC
ba
xba≠+
−
−
( ) ( )( )
−−
−−=∫ ba
xbadxbxax
2
sinsinsin
( )( )
( )22
2
sinbaC
ba
xba≠+
+
+
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( ) ( )∫ +−=
−
na
axaxdxax
nn cossin
sin1
( ) Cdxaxn
n n +−∫
−2sin1
( )∫ +=−
na
axaxdxax
nn sincos
cos1
Caxn
n n +−∫
−2cos1
( )∫ =dxaxntan
( )( ) ( )1tan
1
tan 21
≠+−− ∫
−−
nCdxaxna
ax nn
( ) ( )( )∫ +
−=
−
1
cotcsccsc
2
na
axaxdxax
nn
( ) ( )1csc1
2 2 ≠+−
−∫
−nCdxax
n
n n
( ) ( )( )∫ +
−=
−
1
tansecsec
2
na
axaxdxax
nn
( ) ( )1sec1
2 2 ≠+−
−∫
−nCdxax
n
n n
( )( )∫ −
−−=
−
1
cotcot
1
na
axdxax
nn
( ) ( )1cot 2 ≠+∫− nCdxaxn
( ) =∫− dxax1sin
Cxaa
axx +−+− 221 11
sin
Caxaax
dx+−=∫ cot
1
sin 2
Caxaax
dx+=∫ tan
1
cos2
∫ +
+=
−C
ax
aax
dx
24tan
1
sin1
π
(x in rads)
Cax
aax
dx+
−−=
+∫ 24tan
1
sin1
π
(x in rads)
∫ +−=−
Cax
aax
dx
2cot
1
cos1
∫ +=+
Cax
aax
dx
2tan
1
cos1
∫ =+ axcb
dx
sin
−
+
−
−
− −
24tantan
2 1
22
ax
cb
cb
cba
π
( )22 cbC >+
( )∫ =dxaxx sin2
Caxa
xaax
a
x+
+− cos
2sin
23
22
2
( ) =∫ dxaxx cos2
Caxa
xaax
a
x+
−+ sin
2cos
23
22
2
( )∫ =dxaxx n sin
( ) Cdxaxxa
nax
a
x nn
++− ∫− coscos 1
( ) =∫ dxaxx n cos
( ) Cdxaxxa
nax
a
x n +− ∫− sinsin 1
2
( )∫ =− dxaxx n 1sin
Cxa
dxx
n
aax
n
xnn
+−+
−+ ∫
+−
+
22
11
1
11sin
1
( )1−≠n
( )∫ ++
= −+
−ax
n
xdxaxx
nn 1
11 cos
1cos
( )111 22
1
−≠+−+ ∫
+
nCxa
dxx
n
an
( )∫ −+
= −−
−ax
n
xdxaxx
nn 1
11 tan
1tan
( )111 22
1
−≠+−+ ∫
+
nCxa
dxx
n
an
( )∫ +−= Cxdxxx csccotcsc
( )∫ += Cxdxxx sectansec
( )∫ +−= Ca
axdxaxax
4
2coscossin
( )∫ += Caxa
dxax sinh1
cosh
( ) ( )∫ += Caxa
dxax coshln1
tanh
( )∫ += Cax
adxhax
2tanhln
1csc
( ) ( )∫ += −Cax
adxhax tanhsin
1sec 1
( )∫ += Caxa
dxax sinhln1
coth
( )∫ +−= Cx
a
axdxax
24
2sinhsinh 2
( )∫ ++= Cx
a
axdxax
24
2sinhcosh 2
( ) ( )( )∫ +
−
−=
ba
xbadxbxax
2
sincoscos
( )( )
( )22
2
sinbaC
ba
xba≠+
+
+
( )( )∫ +
+=
+
Can
axdxaxax
nn
1
sincossin
1
( )1−≠n
( )( )∫ +
+−=
+
Can
axdxaxax
nn
1
cossincos
1
( )1−≠n
( )∫ +−= Cna
axdxaxax
nn csc
cotcsc
( )0≠n
( ) Cna
axdxaxax
nn +=∫
sectansec
( )0≠n
( ) =∫ dxaxax mn cossin
( )
( )+
+−
+−
nma
axaxmn 11 cossin
( )∫ ++
− −Cdxaxax
nm
n mn cossin1 2
( )mn −≠
( ) =∫ dxaxax mn cossin
( )
( )+
+
−+
nma
axaxmn 11 cossin
( )∫ ++
+ −Cdxaxax
nm
m mn 2cossin1
( )nm −≠
( )∫ =dxbxeax sin
( ) Cbxbbxaba
eax
+−+
cossin22
( )∫ =dxbxeax cos
( ) Cbxbbxaba
eax
+−+
sincos22
( )∫ =dxaxx sinh
Caxa
axx
a+− sinh
1cosh
2
( )∫ =dxaxx cosh
Caxa
axx
a+− cosh
1sinh
2
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∫ =+ axcb
dx
sin
axcb
axbcaxbc
bca sin
cossinln
1 22
22 +
−++
−
−
( )22 bcC >+
∫ =+ axcb
dx
cos
Cax
cb
cb
cba+
+
−
−
−
2tantan
2 1
22
( )22 cb >
∫ =+ axcb
dx
cos
Caxcb
axbcaxbc
bca+
+
−++
− cos
sincosln
1 22
22
( )22 cb <
( ) ( )∫ += Cxdxx coshsinh
( ) ( )∫ += Cxdxx sinhcosh
( ) ( )∫ += Cxdxx coshlntanh
( )∫ += Cx
dxhx2
tanhlncsc
( ) ( )∫ += − Cxdxhx tanhsinsec 1
( )∫ += Cxdxx sinhlncoth
( )∫ += Caxa
dxax cosh1
sinh
( )∫ +−= Caxa
xdxax tanh1
tanh 2
( )∫ +−= Caxa
dxaxh coth1
csc 2
( )∫ += Caxa
dxaxh tanh1
sec 2
( )∫ +−= Caxa
xdxax coth1
coth 2
( ) ( )∫ −=
−
na
axaxdxax
nn cossinh
sinh1
( ) ( )0sinh1 2 ≠+
−∫
−nCdxax
n
n n
( ) ( )∫ +=
−
na
axaxdxax
nn sincosh
cosh1
( ) ( )0cosh1 2 ≠+
−∫
−nCdxax
n
n n
( ) Cdxaxn +=∫ tanh
( )
( ) Cdxaxan
ax nn
++− ∫
−−
21
tanh1
tanh
( )1≠n
( )( )∫ +
−−=
−
an
axaxhdxaxh
nn
1
cothcsccsc
2
( ) ( )1csc1
2 2 ≠+−
−∫
−nCdxaxh
n
n n
( )( )∫ +
−=
−
an
axaxhdxaxh
nn
1
tanhsecsec
2
( ) ( )1sec1
2 2 ≠+−
−∫
−nCdxaxh
n
n n
( )( )∫ +
−=
−
an
axdxax
nn
1
cothcoth
1
( ) ( )1coth 2 ≠+∫− nCdxaxn
( )∫ =dxaxx n sinh
( ) Cdxaxxa
nax
a
x nn
+− ∫− coshcosh 1
( )∫ =dxaxx n cosh
( ) Cdxaxxa
nax
a
x nn
+− ∫− sinhsinh 1
( ) Cna
axhdxaxaxh
nn +−=∫
sectanhsec
( )0≠n
( ) Cna
axhdxaxaxh
nn +−=∫
csccothcsc
( )0≠n
( )∫ =dxbxe ax sinh
( )22
2baC
ba
e
ba
ee bxbxax
≠+
−−
+
−
( )∫ =dxbxe ax cosh
( )22
2baC
ba
e
ba
ee bxbxax
≠+
−+
+
−