Home » Archimedes archive » Acorn User » AU 1997-01 B.adf » Regulars » StarInfo/Allen/!Ignotum/Formulae/Formulae/Inte
StarInfo/Allen/!Ignotum/Formulae/Formulae/Inte
This website contains an archive of files for the Acorn Electron, BBC Micro, Acorn Archimedes, Commodore 16 and Commodore 64 computers, which Dominic Ford has rescued from his private collection of floppy disks and cassettes.
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| Tape/disk: | Home » Archimedes archive » Acorn User » AU 1997-01 B.adf » Regulars |
| Filename: | StarInfo/Allen/!Ignotum/Formulae/Formulae/Inte |
| Read OK: | ✔ |
| File size: | 071B bytes |
| Load address: | 0000 |
| Exec address: | 0000 |
File contents
# Maths > Integration
# New entries should take the form of:
# Formula
# Note 1
# Note 2
# Note 3
# Note 4
# Note 5
# Formula
# Note 1
# And so on...
# To fit snugly into the window, each line should be no longer
# than 42 characters.
# There is a limit of 25 formulas per topic.
# Any notes should be made here, at the beginning and should
# be preceeded by a hash (#).
{x^n=x^(n+1)/(n+1)+k
Where:
n does not equal -1
k is a constant of integration
{cosx=sinx+k
Where:
k is a constant of integration
{sinx=-cosx+k
Where:
k is a constant of integration
{tanx=ln|secx|+k
Where:
k is a constant of integration
{cosecx=ln|tan�x|+k
Where:
k is a constant of integration
{secx=ln|tan(45�+�x)|+k
Where:
k is a constant of integration
{cotx=ln|sinx|+k
Where:
k is a constant of integration
{sec�x=tanx+k
Where:
k is a constant of integration
{secxtanx=secx+k
Where:
k is a constant of integration
{cosec�x=-cotx+k
Where:
k is a constant of integration
{1/x=ln|x|+k
Where:
k is a constant of integration
{e^x=e^x+k
Where:
k is a constant of integration
{1/(a�-x�)^�=arcsin(x/a)+k
Where:
k is a constant of integration
Note: arcsinx is also written as
sin^(-1)x
{a/(a�+x�)=arctan(x/a)+k
Where:
k is a constant of integration
Note: arctanx is also written as
tan^(-1)x
{f'(x)/f(x)=ln|f(x)|+k
Where:
f(x) is a function of x
f'(x) is the differentiation of this
function
k is a constant of integration
f(x)=f(g(u))(dx/du)
This is called integration by
substitution, where you replace a
function of x with u, and multiply it by
dx/du.
{u(dv/dx)=uv-{(du/dx)v
This is called integration by parts.
For example, to integrate xe^x:
Let u=x and dv/dx=e^x
So du/dx=1 and v=e^x
So {xe^x dx = xe^x-{e^x = e^x(x-1)+k
00000000 23 20 4d 61 74 68 73 20 3e 20 49 6e 74 65 67 72 |# Maths > Integr|
00000010 61 74 69 6f 6e 0a 0a 23 20 4e 65 77 20 65 6e 74 |ation..# New ent|
00000020 72 69 65 73 20 73 68 6f 75 6c 64 20 74 61 6b 65 |ries should take|
00000030 20 74 68 65 20 66 6f 72 6d 20 6f 66 3a 0a 23 20 | the form of:.# |
00000040 20 20 20 20 46 6f 72 6d 75 6c 61 0a 23 20 20 20 | Formula.# |
00000050 20 20 4e 6f 74 65 20 31 0a 23 20 20 20 20 20 4e | Note 1.# N|
00000060 6f 74 65 20 32 0a 23 20 20 20 20 20 4e 6f 74 65 |ote 2.# Note|
00000070 20 33 0a 23 20 20 20 20 20 4e 6f 74 65 20 34 0a | 3.# Note 4.|
00000080 23 20 20 20 20 20 4e 6f 74 65 20 35 0a 23 20 20 |# Note 5.# |
00000090 20 20 20 46 6f 72 6d 75 6c 61 0a 23 20 20 20 20 | Formula.# |
000000a0 20 4e 6f 74 65 20 31 0a 23 20 20 20 20 20 41 6e | Note 1.# An|
000000b0 64 20 73 6f 20 6f 6e 2e 2e 2e 0a 23 20 54 6f 20 |d so on....# To |
000000c0 66 69 74 20 73 6e 75 67 6c 79 20 69 6e 74 6f 20 |fit snugly into |
000000d0 74 68 65 20 77 69 6e 64 6f 77 2c 20 65 61 63 68 |the window, each|
000000e0 20 6c 69 6e 65 20 73 68 6f 75 6c 64 20 62 65 20 | line should be |
000000f0 6e 6f 20 6c 6f 6e 67 65 72 0a 23 20 74 68 61 6e |no longer.# than|
00000100 20 34 32 20 63 68 61 72 61 63 74 65 72 73 2e 0a | 42 characters..|
00000110 23 20 54 68 65 72 65 20 69 73 20 61 20 6c 69 6d |# There is a lim|
00000120 69 74 20 6f 66 20 32 35 20 66 6f 72 6d 75 6c 61 |it of 25 formula|
00000130 73 20 70 65 72 20 74 6f 70 69 63 2e 0a 0a 23 20 |s per topic...# |
00000140 41 6e 79 20 6e 6f 74 65 73 20 73 68 6f 75 6c 64 |Any notes should|
00000150 20 62 65 20 6d 61 64 65 20 68 65 72 65 2c 20 61 | be made here, a|
00000160 74 20 74 68 65 20 62 65 67 69 6e 6e 69 6e 67 20 |t the beginning |
00000170 61 6e 64 20 73 68 6f 75 6c 64 0a 23 20 62 65 20 |and should.# be |
00000180 70 72 65 63 65 65 64 65 64 20 62 79 20 61 20 68 |preceeded by a h|
00000190 61 73 68 20 28 23 29 2e 0a 0a 7b 78 5e 6e 3d 78 |ash (#)...{x^n=x|
000001a0 5e 28 6e 2b 31 29 2f 28 6e 2b 31 29 2b 6b 0a 57 |^(n+1)/(n+1)+k.W|
000001b0 68 65 72 65 3a 0a 6e 20 64 6f 65 73 20 6e 6f 74 |here:.n does not|
000001c0 20 65 71 75 61 6c 20 2d 31 0a 6b 20 69 73 20 61 | equal -1.k is a|
000001d0 20 63 6f 6e 73 74 61 6e 74 20 6f 66 20 69 6e 74 | constant of int|
000001e0 65 67 72 61 74 69 6f 6e 0a 0a 0a 7b 63 6f 73 78 |egration...{cosx|
000001f0 3d 73 69 6e 78 2b 6b 0a 57 68 65 72 65 3a 0a 6b |=sinx+k.Where:.k|
00000200 20 69 73 20 61 20 63 6f 6e 73 74 61 6e 74 20 6f | is a constant o|
00000210 66 20 69 6e 74 65 67 72 61 74 69 6f 6e 0a 0a 0a |f integration...|
00000220 0a 7b 73 69 6e 78 3d 2d 63 6f 73 78 2b 6b 0a 57 |.{sinx=-cosx+k.W|
00000230 68 65 72 65 3a 0a 6b 20 69 73 20 61 20 63 6f 6e |here:.k is a con|
00000240 73 74 61 6e 74 20 6f 66 20 69 6e 74 65 67 72 61 |stant of integra|
00000250 74 69 6f 6e 0a 0a 0a 0a 7b 74 61 6e 78 3d 6c 6e |tion....{tanx=ln|
00000260 7c 73 65 63 78 7c 2b 6b 0a 57 68 65 72 65 3a 0a ||secx|+k.Where:.|
00000270 6b 20 69 73 20 61 20 63 6f 6e 73 74 61 6e 74 20 |k is a constant |
00000280 6f 66 20 69 6e 74 65 67 72 61 74 69 6f 6e 0a 0a |of integration..|
00000290 0a 0a 7b 63 6f 73 65 63 78 3d 6c 6e 7c 74 61 6e |..{cosecx=ln|tan|
000002a0 bd 78 7c 2b 6b 0a 57 68 65 72 65 3a 0a 6b 20 69 |.x|+k.Where:.k i|
000002b0 73 20 61 20 63 6f 6e 73 74 61 6e 74 20 6f 66 20 |s a constant of |
000002c0 69 6e 74 65 67 72 61 74 69 6f 6e 0a 0a 0a 0a 7b |integration....{|
000002d0 73 65 63 78 3d 6c 6e 7c 74 61 6e 28 34 35 b0 2b |secx=ln|tan(45.+|
000002e0 bd 78 29 7c 2b 6b 0a 57 68 65 72 65 3a 0a 6b 20 |.x)|+k.Where:.k |
000002f0 69 73 20 61 20 63 6f 6e 73 74 61 6e 74 20 6f 66 |is a constant of|
00000300 20 69 6e 74 65 67 72 61 74 69 6f 6e 0a 0a 0a 0a | integration....|
00000310 7b 63 6f 74 78 3d 6c 6e 7c 73 69 6e 78 7c 2b 6b |{cotx=ln|sinx|+k|
00000320 0a 57 68 65 72 65 3a 0a 6b 20 69 73 20 61 20 63 |.Where:.k is a c|
00000330 6f 6e 73 74 61 6e 74 20 6f 66 20 69 6e 74 65 67 |onstant of integ|
00000340 72 61 74 69 6f 6e 0a 0a 0a 0a 7b 73 65 63 b2 78 |ration....{sec.x|
00000350 3d 74 61 6e 78 2b 6b 0a 57 68 65 72 65 3a 0a 6b |=tanx+k.Where:.k|
00000360 20 69 73 20 61 20 63 6f 6e 73 74 61 6e 74 20 6f | is a constant o|
00000370 66 20 69 6e 74 65 67 72 61 74 69 6f 6e 0a 0a 0a |f integration...|
00000380 0a 7b 73 65 63 78 74 61 6e 78 3d 73 65 63 78 2b |.{secxtanx=secx+|
00000390 6b 0a 57 68 65 72 65 3a 0a 6b 20 69 73 20 61 20 |k.Where:.k is a |
000003a0 63 6f 6e 73 74 61 6e 74 20 6f 66 20 69 6e 74 65 |constant of inte|
000003b0 67 72 61 74 69 6f 6e 0a 0a 0a 0a 7b 63 6f 73 65 |gration....{cose|
000003c0 63 b2 78 3d 2d 63 6f 74 78 2b 6b 0a 57 68 65 72 |c.x=-cotx+k.Wher|
000003d0 65 3a 0a 6b 20 69 73 20 61 20 63 6f 6e 73 74 61 |e:.k is a consta|
000003e0 6e 74 20 6f 66 20 69 6e 74 65 67 72 61 74 69 6f |nt of integratio|
000003f0 6e 0a 0a 0a 0a 7b 31 2f 78 3d 6c 6e 7c 78 7c 2b |n....{1/x=ln|x|+|
00000400 6b 0a 57 68 65 72 65 3a 0a 6b 20 69 73 20 61 20 |k.Where:.k is a |
00000410 63 6f 6e 73 74 61 6e 74 20 6f 66 20 69 6e 74 65 |constant of inte|
00000420 67 72 61 74 69 6f 6e 0a 0a 0a 0a 7b 65 5e 78 3d |gration....{e^x=|
00000430 65 5e 78 2b 6b 0a 57 68 65 72 65 3a 0a 6b 20 69 |e^x+k.Where:.k i|
00000440 73 20 61 20 63 6f 6e 73 74 61 6e 74 20 6f 66 20 |s a constant of |
00000450 69 6e 74 65 67 72 61 74 69 6f 6e 0a 0a 0a 0a 7b |integration....{|
00000460 31 2f 28 61 b2 2d 78 b2 29 5e bd 3d 61 72 63 73 |1/(a.-x.)^.=arcs|
00000470 69 6e 28 78 2f 61 29 2b 6b 0a 57 68 65 72 65 3a |in(x/a)+k.Where:|
00000480 0a 6b 20 69 73 20 61 20 63 6f 6e 73 74 61 6e 74 |.k is a constant|
00000490 20 6f 66 20 69 6e 74 65 67 72 61 74 69 6f 6e 0a | of integration.|
000004a0 4e 6f 74 65 3a 20 61 72 63 73 69 6e 78 20 69 73 |Note: arcsinx is|
000004b0 20 61 6c 73 6f 20 77 72 69 74 74 65 6e 20 61 73 | also written as|
000004c0 0a 73 69 6e 5e 28 2d 31 29 78 0a 0a 7b 61 2f 28 |.sin^(-1)x..{a/(|
000004d0 61 b2 2b 78 b2 29 3d 61 72 63 74 61 6e 28 78 2f |a.+x.)=arctan(x/|
000004e0 61 29 2b 6b 0a 57 68 65 72 65 3a 0a 6b 20 69 73 |a)+k.Where:.k is|
000004f0 20 61 20 63 6f 6e 73 74 61 6e 74 20 6f 66 20 69 | a constant of i|
00000500 6e 74 65 67 72 61 74 69 6f 6e 0a 4e 6f 74 65 3a |ntegration.Note:|
00000510 20 61 72 63 74 61 6e 78 20 69 73 20 61 6c 73 6f | arctanx is also|
00000520 20 77 72 69 74 74 65 6e 20 61 73 0a 74 61 6e 5e | written as.tan^|
00000530 28 2d 31 29 78 0a 0a 7b 66 27 28 78 29 2f 66 28 |(-1)x..{f'(x)/f(|
00000540 78 29 3d 6c 6e 7c 66 28 78 29 7c 2b 6b 0a 57 68 |x)=ln|f(x)|+k.Wh|
00000550 65 72 65 3a 0a 66 28 78 29 20 69 73 20 61 20 66 |ere:.f(x) is a f|
00000560 75 6e 63 74 69 6f 6e 20 6f 66 20 78 0a 66 27 28 |unction of x.f'(|
00000570 78 29 20 69 73 20 74 68 65 20 64 69 66 66 65 72 |x) is the differ|
00000580 65 6e 74 69 61 74 69 6f 6e 20 6f 66 20 74 68 69 |entiation of thi|
00000590 73 0a 66 75 6e 63 74 69 6f 6e 0a 6b 20 69 73 20 |s.function.k is |
000005a0 61 20 63 6f 6e 73 74 61 6e 74 20 6f 66 20 69 6e |a constant of in|
000005b0 74 65 67 72 61 74 69 6f 6e 0a 66 28 78 29 3d 66 |tegration.f(x)=f|
000005c0 28 67 28 75 29 29 28 64 78 2f 64 75 29 0a 54 68 |(g(u))(dx/du).Th|
000005d0 69 73 20 69 73 20 63 61 6c 6c 65 64 20 69 6e 74 |is is called int|
000005e0 65 67 72 61 74 69 6f 6e 20 62 79 0a 73 75 62 73 |egration by.subs|
000005f0 74 69 74 75 74 69 6f 6e 2c 20 77 68 65 72 65 20 |titution, where |
00000600 79 6f 75 20 72 65 70 6c 61 63 65 20 61 0a 66 75 |you replace a.fu|
00000610 6e 63 74 69 6f 6e 20 6f 66 20 78 20 77 69 74 68 |nction of x with|
00000620 20 75 2c 20 61 6e 64 20 6d 75 6c 74 69 70 6c 79 | u, and multiply|
00000630 20 69 74 20 62 79 0a 64 78 2f 64 75 2e 0a 0a 7b | it by.dx/du...{|
00000640 75 28 64 76 2f 64 78 29 3d 75 76 2d 7b 28 64 75 |u(dv/dx)=uv-{(du|
00000650 2f 64 78 29 76 0a 54 68 69 73 20 69 73 20 63 61 |/dx)v.This is ca|
00000660 6c 6c 65 64 20 69 6e 74 65 67 72 61 74 69 6f 6e |lled integration|
00000670 20 62 79 20 70 61 72 74 73 2e 0a 46 6f 72 20 65 | by parts..For e|
00000680 78 61 6d 70 6c 65 2c 20 74 6f 20 69 6e 74 65 67 |xample, to integ|
00000690 72 61 74 65 20 78 65 5e 78 3a 0a 4c 65 74 20 75 |rate xe^x:.Let u|
000006a0 3d 78 20 61 6e 64 20 64 76 2f 64 78 3d 65 5e 78 |=x and dv/dx=e^x|
000006b0 0a 53 6f 20 64 75 2f 64 78 3d 31 20 61 6e 64 20 |.So du/dx=1 and |
000006c0 76 3d 65 5e 78 0a 53 6f 20 7b 78 65 5e 78 20 64 |v=e^x.So {xe^x d|
000006d0 78 20 3d 20 78 65 5e 78 2d 7b 65 5e 78 20 3d 20 |x = xe^x-{e^x = |
000006e0 65 5e 78 28 78 2d 31 29 2b 6b 0a 0a 0a 0a 0a 0a |e^x(x-1)+k......|
000006f0 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a |................|
*
00000710 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a |...........|
0000071b 
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