, Systems of Units. Some Important Conversion Factors
The most important systems of units are shown in the table below. The mks system is also known as
the International System of Units (abbreviated SI), and the abbreviations sec (instead of s),
gm (instead of g), and nt (instead of N) are also used.
System of units Length Mass Time Force
cgs system centimeter (cm) gram (g) second (s) dyne
mks system meter (m) kilogram (kg) second (s) newton (nt)
Engineering system foot (ft) slug second (s) pound (lb)
1 inch (in.) ⫽ 2.540000 cm 1 foot (ft) ⫽ 12 in. ⫽ 30.480000 cm
1 yard (yd) ⫽ 3 ft ⫽ 91.440000 cm 1 statute mile (mi) ⫽ 5280 ft ⫽ 1.609344 km
1 nautical mile ⫽ 6080 ft ⫽ 1.853184 km
1 acre ⫽ 4840 yd2 ⫽ 4046.8564 m2 1 mi2 ⫽ 640 acres ⫽ 2.5899881 km2
1 fluid ounce ⫽ 1/128 U.S. gallon ⫽ 231/128 in.3 ⫽ 29.573730 cm3
1 U.S. gallon ⫽ 4 quarts (liq) ⫽ 8 pints (liq) ⫽ 128 fl oz ⫽ 3785.4118 cm3
1 British Imperial and Canadian gallon ⫽ 1.200949 U.S. gallons ⫽ 4546.087 cm3
1 slug ⫽ 14.59390 kg
1 pound (lb) ⫽ 4.448444 nt 1 newton (nt) ⫽ 105 dynes
1 British thermal unit (Btu) ⫽ 1054.35 joules 1 joule ⫽ 107 ergs
1 calorie (cal) ⫽ 4.1840 joules
1 kilowatt-hour (kWh) ⫽ 3414.4 Btu ⫽ 3.6 • 106 joules
1 horsepower (hp) ⫽ 2542.48 Btu/h ⫽ 178.298 cal/sec ⫽ 0.74570 kW
1 kilowatt (kW) ⫽ 1000 watts ⫽ 3414.43 Btu/h ⫽ 238.662 cal/s
°F ⫽ °C • 1.8 ⫹ 32 1° ⫽ 60⬘ ⫽ 3600⬙ ⫽ 0.017453293 radian
For further details see, for example, D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics. 9th ed., Hoboken,
N. J: Wiley, 2011. See also AN American National Standard, ASTM/IEEE Standard Metric Practice, Institute of Electrical and
Electronics Engineers, Inc. (IEEE), 445 Hoes Lane, Piscataway, N. J. 08854, website at www.ieee.org.
, Differentiation Integration
(cu)⬘ ⫽ cu⬘ (c constant) 冕 uv⬘ dx ⫽ uv ⫺ 冕 u⬘v dx (by parts)
冕 x dx ⫽ nx ⫹ 1 ⫹ c (n ⫽ ⫺1)
n⫹1
n
(u ⫹ v)⬘ ⫽ u⬘ ⫹ v⬘
(uv)⬘ ⫽ u⬘v ⫹ uv⬘ 冕 1x dx ⫽ ln 兩x兩 ⫹ c
u ⬘ u⬘v ⫺ uv⬘
(ᎏ) ⫽ ᎏᎏ 冕 e dx ⫽ 1a e ⫹ c
ax ax
v v2
du du dy
冕 sin x dx ⫽ ⫺cos x ⫹ c
ᎏ⫽ᎏ•ᎏ
冕 cos x dx ⫽ sin x ⫹ c
(Chain rule)
dx dy dx
冕 tan x dx ⫽ ⫺ln 兩cos x兩 ⫹ c
(x n)⬘ ⫽ nxnⴚ1
冕 cot x dx ⫽ ln 兩sin x兩 ⫹ c
(e x)⬘ ⫽ e x
冕 sec x dx ⫽ ln 兩sec x ⫹ tan x兩 ⫹ c
(e ax)⬘ ⫽ ae ax
(a x)⬘ ⫽ a x ln a
冕 csc x dx ⫽ ln 兩csc x ⫺ cot x兩 ⫹ c
(sin x)⬘ ⫽ cos x 冕ᎏ dx
x ⫹a
2
1
a2
x
⫽ ᎏ arctan ᎏ ⫹ c
a
(cos x)⬘ ⫽ ⫺sin x
冕 ᎏᎏ dx
兹a苶苶⫺
苶苶x 苶 2
x
⫽ arcsin ᎏ ⫹ c
2
a
(tan x)⬘ ⫽ sec2 x
(cot x)⬘ ⫽ ⫺csc2 x 冕 ᎏᎏ dx
兹x苶苶⫹苶a苶 2
x
⫽ arcsinh ᎏ ⫹ c
2
a
(sinh x)⬘ ⫽ cosh x
冕 ᎏᎏ dx
兹x苶苶⫺苶a苶 2
x
⫽ arccosh ᎏ ⫹ c
2
a
(cosh x)⬘ ⫽ sinh x
1
冕 sin x dx ⫽ _ x ⫺ _ sin 2x ⫹ c
2 1
2
1
4
(ln x)⬘ ⫽ ᎏ
x 冕 cos x dx ⫽ _ x ⫹ _ sin 2x ⫹ c
2 1
2
1
4
loga e
(loga x)⬘ ⫽ ᎏ 冕 tan x dx ⫽ tan x ⫺ x ⫹ c
2
x
1
冕 cot x dx ⫽ ⫺cot x ⫺ x ⫹ c
2
(arcsin x)⬘ ⫽ ᎏᎏ
兹1苶苶
⫺苶x 2苶 冕 ln x dx ⫽ x ln x ⫺ x ⫹ c
(arccos x)⬘ ⫽ ⫺ ᎏᎏ
1 冕 e sin bx dx
ax
兹1苶苶
⫺苶x 2苶 eax
⫽ (a sin bx ⫺ b cos bx) ⫹ c
1 a2 ⫹ b 2
(arctan x)⬘ ⫽ ᎏ
1 ⫹ x2
冕eax
cos bx dx
1 eax
(arccot x)⬘ ⫽ ⫺ ᎏ ⫽ 2 (a cos bx ⫹ b sin bx) ⫹ c
1 ⫹ x2 a ⫹ b2
,
The most important systems of units are shown in the table below. The mks system is also known as
the International System of Units (abbreviated SI), and the abbreviations sec (instead of s),
gm (instead of g), and nt (instead of N) are also used.
System of units Length Mass Time Force
cgs system centimeter (cm) gram (g) second (s) dyne
mks system meter (m) kilogram (kg) second (s) newton (nt)
Engineering system foot (ft) slug second (s) pound (lb)
1 inch (in.) ⫽ 2.540000 cm 1 foot (ft) ⫽ 12 in. ⫽ 30.480000 cm
1 yard (yd) ⫽ 3 ft ⫽ 91.440000 cm 1 statute mile (mi) ⫽ 5280 ft ⫽ 1.609344 km
1 nautical mile ⫽ 6080 ft ⫽ 1.853184 km
1 acre ⫽ 4840 yd2 ⫽ 4046.8564 m2 1 mi2 ⫽ 640 acres ⫽ 2.5899881 km2
1 fluid ounce ⫽ 1/128 U.S. gallon ⫽ 231/128 in.3 ⫽ 29.573730 cm3
1 U.S. gallon ⫽ 4 quarts (liq) ⫽ 8 pints (liq) ⫽ 128 fl oz ⫽ 3785.4118 cm3
1 British Imperial and Canadian gallon ⫽ 1.200949 U.S. gallons ⫽ 4546.087 cm3
1 slug ⫽ 14.59390 kg
1 pound (lb) ⫽ 4.448444 nt 1 newton (nt) ⫽ 105 dynes
1 British thermal unit (Btu) ⫽ 1054.35 joules 1 joule ⫽ 107 ergs
1 calorie (cal) ⫽ 4.1840 joules
1 kilowatt-hour (kWh) ⫽ 3414.4 Btu ⫽ 3.6 • 106 joules
1 horsepower (hp) ⫽ 2542.48 Btu/h ⫽ 178.298 cal/sec ⫽ 0.74570 kW
1 kilowatt (kW) ⫽ 1000 watts ⫽ 3414.43 Btu/h ⫽ 238.662 cal/s
°F ⫽ °C • 1.8 ⫹ 32 1° ⫽ 60⬘ ⫽ 3600⬙ ⫽ 0.017453293 radian
For further details see, for example, D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics. 9th ed., Hoboken,
N. J: Wiley, 2011. See also AN American National Standard, ASTM/IEEE Standard Metric Practice, Institute of Electrical and
Electronics Engineers, Inc. (IEEE), 445 Hoes Lane, Piscataway, N. J. 08854, website at www.ieee.org.
, Differentiation Integration
(cu)⬘ ⫽ cu⬘ (c constant) 冕 uv⬘ dx ⫽ uv ⫺ 冕 u⬘v dx (by parts)
冕 x dx ⫽ nx ⫹ 1 ⫹ c (n ⫽ ⫺1)
n⫹1
n
(u ⫹ v)⬘ ⫽ u⬘ ⫹ v⬘
(uv)⬘ ⫽ u⬘v ⫹ uv⬘ 冕 1x dx ⫽ ln 兩x兩 ⫹ c
u ⬘ u⬘v ⫺ uv⬘
(ᎏ) ⫽ ᎏᎏ 冕 e dx ⫽ 1a e ⫹ c
ax ax
v v2
du du dy
冕 sin x dx ⫽ ⫺cos x ⫹ c
ᎏ⫽ᎏ•ᎏ
冕 cos x dx ⫽ sin x ⫹ c
(Chain rule)
dx dy dx
冕 tan x dx ⫽ ⫺ln 兩cos x兩 ⫹ c
(x n)⬘ ⫽ nxnⴚ1
冕 cot x dx ⫽ ln 兩sin x兩 ⫹ c
(e x)⬘ ⫽ e x
冕 sec x dx ⫽ ln 兩sec x ⫹ tan x兩 ⫹ c
(e ax)⬘ ⫽ ae ax
(a x)⬘ ⫽ a x ln a
冕 csc x dx ⫽ ln 兩csc x ⫺ cot x兩 ⫹ c
(sin x)⬘ ⫽ cos x 冕ᎏ dx
x ⫹a
2
1
a2
x
⫽ ᎏ arctan ᎏ ⫹ c
a
(cos x)⬘ ⫽ ⫺sin x
冕 ᎏᎏ dx
兹a苶苶⫺
苶苶x 苶 2
x
⫽ arcsin ᎏ ⫹ c
2
a
(tan x)⬘ ⫽ sec2 x
(cot x)⬘ ⫽ ⫺csc2 x 冕 ᎏᎏ dx
兹x苶苶⫹苶a苶 2
x
⫽ arcsinh ᎏ ⫹ c
2
a
(sinh x)⬘ ⫽ cosh x
冕 ᎏᎏ dx
兹x苶苶⫺苶a苶 2
x
⫽ arccosh ᎏ ⫹ c
2
a
(cosh x)⬘ ⫽ sinh x
1
冕 sin x dx ⫽ _ x ⫺ _ sin 2x ⫹ c
2 1
2
1
4
(ln x)⬘ ⫽ ᎏ
x 冕 cos x dx ⫽ _ x ⫹ _ sin 2x ⫹ c
2 1
2
1
4
loga e
(loga x)⬘ ⫽ ᎏ 冕 tan x dx ⫽ tan x ⫺ x ⫹ c
2
x
1
冕 cot x dx ⫽ ⫺cot x ⫺ x ⫹ c
2
(arcsin x)⬘ ⫽ ᎏᎏ
兹1苶苶
⫺苶x 2苶 冕 ln x dx ⫽ x ln x ⫺ x ⫹ c
(arccos x)⬘ ⫽ ⫺ ᎏᎏ
1 冕 e sin bx dx
ax
兹1苶苶
⫺苶x 2苶 eax
⫽ (a sin bx ⫺ b cos bx) ⫹ c
1 a2 ⫹ b 2
(arctan x)⬘ ⫽ ᎏ
1 ⫹ x2
冕eax
cos bx dx
1 eax
(arccot x)⬘ ⫽ ⫺ ᎏ ⫽ 2 (a cos bx ⫹ b sin bx) ⫹ c
1 ⫹ x2 a ⫹ b2
,
