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!Ignotum/Formulae/Inte
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| Tape/disk: | Home » Archimedes archive » Archimedes World » AW-1996-07.adf » !Ignotum_Ignotum |
| Filename: | !Ignotum/Formulae/Inte |
| Read OK: | ✔ |
| File size: | 0597 bytes |
| Load address: | 0000 |
| Exec address: | 0000 |
File contents
# Maths > Integration
{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 7b 78 5e 6e 3d 78 5e 28 6e 2b |ation.{x^n=x^(n+|
00000020 31 29 2f 28 6e 2b 31 29 2b 6b 0a 57 68 65 72 65 |1)/(n+1)+k.Where|
00000030 3a 0a 6e 20 64 6f 65 73 20 6e 6f 74 20 65 71 75 |:.n does not equ|
00000040 61 6c 20 2d 31 0a 6b 20 69 73 20 61 20 63 6f 6e |al -1.k is a con|
00000050 73 74 61 6e 74 20 6f 66 20 69 6e 74 65 67 72 61 |stant of integra|
00000060 74 69 6f 6e 0a 0a 0a 7b 63 6f 73 78 3d 73 69 6e |tion...{cosx=sin|
00000070 78 2b 6b 0a 57 68 65 72 65 3a 0a 6b 20 69 73 20 |x+k.Where:.k is |
00000080 61 20 63 6f 6e 73 74 61 6e 74 20 6f 66 20 69 6e |a constant of in|
00000090 74 65 67 72 61 74 69 6f 6e 0a 0a 0a 0a 7b 73 69 |tegration....{si|
000000a0 6e 78 3d 2d 63 6f 73 78 2b 6b 0a 57 68 65 72 65 |nx=-cosx+k.Where|
000000b0 3a 0a 6b 20 69 73 20 61 20 63 6f 6e 73 74 61 6e |:.k is a constan|
000000c0 74 20 6f 66 20 69 6e 74 65 67 72 61 74 69 6f 6e |t of integration|
000000d0 0a 0a 0a 0a 7b 74 61 6e 78 3d 6c 6e 7c 73 65 63 |....{tanx=ln|sec|
000000e0 78 7c 2b 6b 0a 57 68 65 72 65 3a 0a 6b 20 69 73 |x|+k.Where:.k is|
000000f0 20 61 20 63 6f 6e 73 74 61 6e 74 20 6f 66 20 69 | a constant of i|
00000100 6e 74 65 67 72 61 74 69 6f 6e 0a 0a 0a 0a 7b 63 |ntegration....{c|
00000110 6f 73 65 63 78 3d 6c 6e 7c 74 61 6e bd 78 7c 2b |osecx=ln|tan.x|+|
00000120 6b 0a 57 68 65 72 65 3a 0a 6b 20 69 73 20 61 20 |k.Where:.k is a |
00000130 63 6f 6e 73 74 61 6e 74 20 6f 66 20 69 6e 74 65 |constant of inte|
00000140 67 72 61 74 69 6f 6e 0a 0a 0a 0a 7b 73 65 63 78 |gration....{secx|
00000150 3d 6c 6e 7c 74 61 6e 28 34 35 b0 2b bd 78 29 7c |=ln|tan(45.+.x)||
00000160 2b 6b 0a 57 68 65 72 65 3a 0a 6b 20 69 73 20 61 |+k.Where:.k is a|
00000170 20 63 6f 6e 73 74 61 6e 74 20 6f 66 20 69 6e 74 | constant of int|
00000180 65 67 72 61 74 69 6f 6e 0a 0a 0a 0a 7b 63 6f 74 |egration....{cot|
00000190 78 3d 6c 6e 7c 73 69 6e 78 7c 2b 6b 0a 57 68 65 |x=ln|sinx|+k.Whe|
000001a0 72 65 3a 0a 6b 20 69 73 20 61 20 63 6f 6e 73 74 |re:.k is a const|
000001b0 61 6e 74 20 6f 66 20 69 6e 74 65 67 72 61 74 69 |ant of integrati|
000001c0 6f 6e 0a 0a 0a 0a 7b 73 65 63 b2 78 3d 74 61 6e |on....{sec.x=tan|
000001d0 78 2b 6b 0a 57 68 65 72 65 3a 0a 6b 20 69 73 20 |x+k.Where:.k is |
000001e0 61 20 63 6f 6e 73 74 61 6e 74 20 6f 66 20 69 6e |a constant of in|
000001f0 74 65 67 72 61 74 69 6f 6e 0a 0a 0a 0a 7b 73 65 |tegration....{se|
00000200 63 78 74 61 6e 78 3d 73 65 63 78 2b 6b 0a 57 68 |cxtanx=secx+k.Wh|
00000210 65 72 65 3a 0a 6b 20 69 73 20 61 20 63 6f 6e 73 |ere:.k is a cons|
00000220 74 61 6e 74 20 6f 66 20 69 6e 74 65 67 72 61 74 |tant of integrat|
00000230 69 6f 6e 0a 0a 0a 0a 7b 63 6f 73 65 63 b2 78 3d |ion....{cosec.x=|
00000240 2d 63 6f 74 78 2b 6b 0a 57 68 65 72 65 3a 0a 6b |-cotx+k.Where:.k|
00000250 20 69 73 20 61 20 63 6f 6e 73 74 61 6e 74 20 6f | is a constant o|
00000260 66 20 69 6e 74 65 67 72 61 74 69 6f 6e 0a 0a 0a |f integration...|
00000270 0a 7b 31 2f 78 3d 6c 6e 7c 78 7c 2b 6b 0a 57 68 |.{1/x=ln|x|+k.Wh|
00000280 65 72 65 3a 0a 6b 20 69 73 20 61 20 63 6f 6e 73 |ere:.k is a cons|
00000290 74 61 6e 74 20 6f 66 20 69 6e 74 65 67 72 61 74 |tant of integrat|
000002a0 69 6f 6e 0a 0a 0a 0a 7b 65 5e 78 3d 65 5e 78 2b |ion....{e^x=e^x+|
000002b0 6b 0a 57 68 65 72 65 3a 0a 6b 20 69 73 20 61 20 |k.Where:.k is a |
000002c0 63 6f 6e 73 74 61 6e 74 20 6f 66 20 69 6e 74 65 |constant of inte|
000002d0 67 72 61 74 69 6f 6e 0a 0a 0a 0a 7b 31 2f 28 61 |gration....{1/(a|
000002e0 b2 2d 78 b2 29 5e bd 3d 61 72 63 73 69 6e 28 78 |.-x.)^.=arcsin(x|
000002f0 2f 61 29 2b 6b 0a 57 68 65 72 65 3a 0a 6b 20 69 |/a)+k.Where:.k i|
00000300 73 20 61 20 63 6f 6e 73 74 61 6e 74 20 6f 66 20 |s a constant of |
00000310 69 6e 74 65 67 72 61 74 69 6f 6e 0a 4e 6f 74 65 |integration.Note|
00000320 3a 20 61 72 63 73 69 6e 78 20 69 73 20 61 6c 73 |: arcsinx is als|
00000330 6f 20 77 72 69 74 74 65 6e 20 61 73 0a 73 69 6e |o written as.sin|
00000340 5e 28 2d 31 29 78 0a 0a 7b 61 2f 28 61 b2 2b 78 |^(-1)x..{a/(a.+x|
00000350 b2 29 3d 61 72 63 74 61 6e 28 78 2f 61 29 2b 6b |.)=arctan(x/a)+k|
00000360 0a 57 68 65 72 65 3a 0a 6b 20 69 73 20 61 20 63 |.Where:.k is a c|
00000370 6f 6e 73 74 61 6e 74 20 6f 66 20 69 6e 74 65 67 |onstant of integ|
00000380 72 61 74 69 6f 6e 0a 4e 6f 74 65 3a 20 61 72 63 |ration.Note: arc|
00000390 74 61 6e 78 20 69 73 20 61 6c 73 6f 20 77 72 69 |tanx is also wri|
000003a0 74 74 65 6e 20 61 73 0a 74 61 6e 5e 28 2d 31 29 |tten as.tan^(-1)|
000003b0 78 0a 0a 7b 66 27 28 78 29 2f 66 28 78 29 3d 6c |x..{f'(x)/f(x)=l|
000003c0 6e 7c 66 28 78 29 7c 2b 6b 0a 57 68 65 72 65 3a |n|f(x)|+k.Where:|
000003d0 0a 66 28 78 29 20 69 73 20 61 20 66 75 6e 63 74 |.f(x) is a funct|
000003e0 69 6f 6e 20 6f 66 20 78 0a 66 27 28 78 29 20 69 |ion of x.f'(x) i|
000003f0 73 20 74 68 65 20 64 69 66 66 65 72 65 6e 74 69 |s the differenti|
00000400 61 74 69 6f 6e 20 6f 66 20 74 68 69 73 0a 66 75 |ation of this.fu|
00000410 6e 63 74 69 6f 6e 0a 6b 20 69 73 20 61 20 63 6f |nction.k is a co|
00000420 6e 73 74 61 6e 74 20 6f 66 20 69 6e 74 65 67 72 |nstant of integr|
00000430 61 74 69 6f 6e 0a 66 28 78 29 3d 66 28 67 28 75 |ation.f(x)=f(g(u|
00000440 29 29 28 64 78 2f 64 75 29 0a 54 68 69 73 20 69 |))(dx/du).This i|
00000450 73 20 63 61 6c 6c 65 64 20 69 6e 74 65 67 72 61 |s called integra|
00000460 74 69 6f 6e 20 62 79 0a 73 75 62 73 74 69 74 75 |tion by.substitu|
00000470 74 69 6f 6e 2c 20 77 68 65 72 65 20 79 6f 75 20 |tion, where you |
00000480 72 65 70 6c 61 63 65 20 61 0a 66 75 6e 63 74 69 |replace a.functi|
00000490 6f 6e 20 6f 66 20 78 20 77 69 74 68 20 75 2c 20 |on of x with u, |
000004a0 61 6e 64 20 6d 75 6c 74 69 70 6c 79 20 69 74 20 |and multiply it |
000004b0 62 79 0a 64 78 2f 64 75 2e 0a 0a 7b 75 28 64 76 |by.dx/du...{u(dv|
000004c0 2f 64 78 29 3d 75 76 2d 7b 28 64 75 2f 64 78 29 |/dx)=uv-{(du/dx)|
000004d0 76 0a 54 68 69 73 20 69 73 20 63 61 6c 6c 65 64 |v.This is called|
000004e0 20 69 6e 74 65 67 72 61 74 69 6f 6e 20 62 79 20 | integration by |
000004f0 70 61 72 74 73 2e 0a 46 6f 72 20 65 78 61 6d 70 |parts..For examp|
00000500 6c 65 2c 20 74 6f 20 69 6e 74 65 67 72 61 74 65 |le, to integrate|
00000510 20 78 65 5e 78 3a 0a 4c 65 74 20 75 3d 78 20 61 | xe^x:.Let u=x a|
00000520 6e 64 20 64 76 2f 64 78 3d 65 5e 78 0a 53 6f 20 |nd dv/dx=e^x.So |
00000530 64 75 2f 64 78 3d 31 20 61 6e 64 20 76 3d 65 5e |du/dx=1 and v=e^|
00000540 78 0a 53 6f 20 7b 78 65 5e 78 20 64 78 20 3d 20 |x.So {xe^x dx = |
00000550 78 65 5e 78 2d 7b 65 5e 78 20 3d 20 65 5e 78 28 |xe^x-{e^x = e^x(|
00000560 78 2d 31 29 2b 6b 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a |x-1)+k..........|
00000570 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a |................|
*
00000590 0a 0a 0a 0a 0a 0a 0a |.......|
00000597 
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