PK``lXñB–Hmimetypetext/x-wxmathmlPK``lXQdBV55 format.txt This file contains a wxMaxima session in the .wxmx format. .wxmx files are .xml-based files contained in a .zip container like .odt or .docx files. After changing their name to end in .zip the .xml and eventual bitmap files inside them can be extracted using any .zip file viewer. The reason why part of a .wxmx file still might still seem to make sense in a ordinary text viewer is that the text portion of .wxmx by default isn't compressed: The text is typically small and compressing it would mean that changing a single character would (with a high probability) change big parts of the whole contents of the compressed .zip archive. Even if version control tools like git and svn that remember all changes that were ever made to a file can handle binary files compression would make the changed part of the file bigger and therefore seriously reduce the efficiency of version control wxMaxima can be downloaded from https://github.com/wxMaxima-developers/wxmaxima. It also is part of the windows installer for maxima (https://wxmaxima-developers.github.io/wxmaxima/). If a .wxmx file is broken but the content.xml portion of the file can still be viewed using an text editor just save the xml's text as "content.xml" and try to open it using a recent version of wxMaxima. If it is valid XML (the XML header is intact, all opened tags are closed again, the text is saved with the text encoding "UTF8 without BOM" and the few special characters XML requires this for are properly escaped) chances are high that wxMaxima will be able to recover all code and text from the XML file. PK``lX)ÈD£ü÷ü÷ content.xml Kedvenc Matematikai Kísérleteim Alapok Halmazok Számok Határérték Differenciálszámítás Integrálszámítás Primitív függvények Határozott integrál Definíció Tétel Tétel: az integrál egyértelműsége Tétel: komplex értékű függvények integrálja Tétel: az integrál linearitása Tétel: az integrál nemnegativitása Következmény: az integrál monotonitása Tétel: Cauchy-kritérium Segédtétel Tétel: az integrál intervallum-additivitása Tétel Definíció integrate(f(x),x,a,b); (%o337) abf

x

dx
integrate(f(x),x,1,-1); (%o338) −−11f

x

dx
Nullahalmazok Lebesgue-feltétel Következmény Tétel Megjegyzés Az integrálfüggvény Tétel Következmény Newton-Leibniz-formula Parciális integrálás határozott integrálokra Helyettesítéses integrálás határozott integrálokra Tétel: improprius integrálás Példák (1) integrate(x^α,x,0,1); Is α positive, negative or zero? Maxima encountered a Lisp error: SIMPLE−ERROR: Console interrupt.Automatically continuing.To enable the Lisp debugger set *debugger−hook* to nil. assume(α<0); (%o184) [α<0] integrate(x^α,x,0,1); Is α equal to −1? Maxima encountered a Lisp error: SIMPLE−ERROR: Console interrupt.Automatically continuing.To enable the Lisp debugger set *debugger−hook* to nil. assume(α>-1); (%o186) [α>−1] integrate(x^α,x,0,1); (%o187) 1α+1 (2) Tétel: összehasonlító teszt Következmény: határérték teszt Megjegyzés Abszolút integrálható függvények Tétel Tétel: parciális integrálás integrálfüggvényekre Tétel Példa 'integrate(1/(1+sin(x)),x,0,%pi); (%o196) 0%pi1sin

x

+1
dx
changevar(%,t=tan(x/2),t,x); solve: using arc−trig functions to get a solution.Some solutions will be lost.(%o197) 2*0inf1

t2+1

*sin

2*atan

t

+t2+1
dt
Improprius integrál Példák forget(α>-1); (%o201) [α>−1] assume(α<-1); (%o202) [α<−1] integrate(x^α,x,1,inf); (%o203) −1α+1 További improprius integrálok Tétel Tétel Feladat (1) x*abs(x)/2+c; (%o205) x*x2+c x*abs(x)/2+c; (%o218) x*x2+c ahol 2>=c>=-9/2; subst(3,x,%)-subst(-2,x,%); (%o219) 132 (2) Primitív függvény nincs. abs(x)+c; (%o210) x+c ahol 2>=c>=0; subst(3,x,%)-subst(-2,x,%); (%o211) 1 (3) x+x^2*abs(x)/3+c; (%o212) x2*x3+x+c x+x^2*abs(x)/3+c; (%o220) x2*x3+x+c ahol 0>=c>=-12; subst(3,x,%)-subst(-2,x,%); (%o221) 343 Feladat (1) integrate(signum(t),t,-2,x); (%o16) −2xsignum

t

dt
(2) Nincs, nem folytonos. (3) integrate(1/2/sqrt(t),t,0,x); Is x positive, negative or zero? Maxima encountered a Lisp error: SIMPLE−ERROR: Console interrupt.Automatically continuing.To enable the Lisp debugger set *debugger−hook* to nil. assume(x>0); (%o18) [x>0] integrate(1/2/sqrt(t),t,0,x); (%o20) x Feladat Igen, például a Riemann-függvény, stb. Feladat (1/1) integrate(sqrt(x)*exp(sqrt(x)),x,0,1); (%o21) 2*%e−4 (1/2) integrate(sqrt(log(x))/x,x,2,3); (%o22) 2*log

3

32
3
−2*log

2

32
3
(1/3) integrate(sin(sqrt(x)),x,0,%pi^2); (%o23) 2*%pi (2/1) integrate(atan(x),x,0,1); (%o24) −2*log

2

−%pi
4
(2/2) integrate(atan(sqrt(x)),x,0,1); (%o25) %pi−22 (2/3) integrate(log(1+x^2),x,0,1); (%o26) 2*log

2

+%pi−4
2
(3/1) integrate(sqrt(x^3+x^2),x,0,1); (%o27) 25215+415 (3/2) integrate(exp(a*x)*cos(b*x),x,0,1); (%o28) %ea*b*sin

b

+a*%ea*cos

b

b2+a2
−ab2+a2
(3/3) integrate(exp(a*c)*sin(b*x),x,0,1); (%o29)

1b−cos

b

b

*%ea*c
Feladat (1/1) integrate(1/sqrt(4-x^2),x,0,2); (%o30) %pi2 (1/2) integrate(1/(x+2)^(4/5),x,-3,0); (%o31) 5*215+5 (1/3) integrate(2^(1/3)*x^(-1/3),x,0,2); (%o32) 3 (2/1) integrate(x^2*exp(-x),x,0,inf); (%o3) 2 (2/2) integrate(x^(-2/3),x,-1,1); (%o33) 6 (2/3) integrate(3*x^2/(x^3-1)^(2/7),x,0,2^(1/3)); (%o34) 145 (2/4) integrate(1/sqrt(2*x+1),x,-1/2,1); (%o36) 3 (3/1) integrate(exp(x)*sqrt(exp(x)-1)/(exp(x)+3),x,0,log(5)); (%o2) 4−%pi (3/2) integrate(1/(x+2),x,-2,0); defint: integral is divergent. −− an error. To debug this try: debugmode(true); (3/3) integrate(asin(sqrt(x))/sqrt(1-x),x,0,1); (%o38) 2 (3/4) integrate(log(x)^2,x,0,1); (%o39) 2 (4/1) integrate(exp(-x),x,0,inf); (%o40) 1 (4/2) integrate(1/sqrt(1+x^2),x,minf,inf); defint: integral is divergent. −− an error. To debug this try: debugmode(true); (4/3) integrate(1/(3+4*x^2),x,minf,1); (%o42) atan

23

2*3
+%pi4*3
(4/4) integrate(1/(1+x^3),x,0,inf); (%o2) 2*%pi332 (5/1) integrate(1/(2*x+1)^(2/3),x,0,inf); defint: integral is divergent. −− an error. To debug this try: debugmode(true); (5/2) integrate(exp(2*x),x,minf,0); (%o45) 12 (5/3) integrate(x*exp(-x),x,0,inf); (%o46) 1 (6/1) integrate(1/(x*log(x)),x,2,inf); defint: integral is divergent. −− an error. To debug this try: debugmode(true); (6/2) integrate(1/(x*log(x)^2),x,2,inf); (%o48) 1log

2

(6/3) integrate(exp(-x)*cos(x),x,0,inf); (%o49) 12 (7/1) integrate((log(x)/x)^2,x,1,inf); (%o50) 2 (7/2) integrate(x^2*exp(-x),x,0,inf); (%o51) 2 (7/3) integrate(1/x^2,x,-1,1); defint: integral is divergent. −− an error. To debug this try: debugmode(true); (8/1) integrate(16*x-x*4^x,x,0,2); (%o53) 128*log

2

2
−64*log

2

+15
4*log

2

2
(8/2) integrate(cos(x)^2,x,0,%pi); (%o54) %pi2 (8/3) integrate(1/(2+3*x^2),x,minf,inf); (%o55) %pi2*3 (9/1) integrate(1/(1+x),x,0,inf); defint: integral is divergent. −− an error. To debug this try: debugmode(true); (9/2) integrate(1/(x-1)^(3/2),x,1,2); defint: integral is divergent. −− an error. To debug this try: debugmode(true); (9/3) integrate(1/sqrt(1-x),x,0,1); (%o58) 2 (10/1) integrate(1/sqrt(1-4*x^2),x,-1/2,0); (%o61) %pi4 (10/2) integrate(1/(x-1)^(2/3),x,0,2); (%o63) 6 (10/3) integrate(1/sqrt(abs(x-2)*abs(x-3)),x,1,4); (%o64) 141x−3*x−2dx integrate(1/sqrt((x-2)(x-3)),x,3,4); (%o66) 341

x−2

x−3

dx
integrate(1/sqrt(x^2-5*x+6),x,3,4); (%o71) log

3*252+17

2
Stb. (11/1) integrate(1/(x^2+x),x,1,inf); (%o73) log

2

(11/2) integrate(1/(x^2+x+1),x,minf,inf); (%o74) 2*%pi3 (11/3) integrate(1/(1+x^4),x,1,4); (%o75) log

252+17

252
+atan

252+1

232
+atan

252−1

232
−log

17−252

252
−log

2+2

252
+log

2−2

252
−%pi252
(12/1) integrate((x+1)/(x^3+x),x,1,inf); (%o76) 2*log

2

−%pi
4
+%pi2
(12/2) integrate(x/(1+x^2)^3,x,0,inf); (%o77) 14 (12/3) integrate(1/sqrt((x-a)*(x-b)),x,a,b); Is b−a positive, negative or zero? Maxima encountered a Lisp error: SIMPLE−ERROR: Console interrupt.Automatically continuing.To enable the Lisp debugger set *debugger−hook* to nil. assume(b-a>0); (%o79) [b>a] integrate(1/sqrt((x-a)*(x-b)),x,a,b); Is b+a zero or nonzero? Maxima encountered a Lisp error: SIMPLE−ERROR: Console interrupt.Automatically continuing.To enable the Lisp debugger set *debugger−hook* to nil. assume(b+a>0); (%o82) [b+a>0] integrate(1/sqrt((x-a)*(x-b)),x,a,b); (%o83) −2*%i*asin

b2−2*a*b+a2b−a

Az integrandus mindenütt negatív, valósban nincs megoldás -%i*integrate(1/sqrt((x-a)*(b-x)),x,a,b); (%o61) −2*%i*asin

b2−2*a*b+a2b−a

radcan(%); (%o62) −%i*%pi (13/1) integrate(1/(2^x-1),x,1,inf); (%o96) 1 (13/2) integrate(exp(x)/(exp(2*x)+1),x,minf,inf); (%o97) %pi2 (14/3) integrate(exp(-sqrt(x)),x,0,inf); (%o98) 2 (15/1) integrate(log(x)/x^3,x,1,inf); (%o99) 14 (15/2) integrate(x*log(x)/(1+x^2)^2,x,1,inf); (%o100) log

2

4
(15/3) integrate(1/sin(x)^4,x,%pi/4,%pi/2); (%o101) 43 Feladat (1/1) limit((x-1)!/(2^x+3^x),x,inf); (%o67) inf (1/2) integrate(1/x/log(log(x)),x,10,inf); defint: integral is divergent. −− an error. To debug this try: debugmode(true); (1/3) integrate(log(abs(log(x))),x,0,1); (%o69) 01log

log

x

dx
integrate(log(-log(x)),x,0,1); (%o103) 01log

−log

x

dx
changevar(%,-log(x)=y,y,x); (%o104) 0inf%e−y*log

y

dy
Konvergens a végtelenben, így az eredeti a nullában, sőt: ev(%o104,nouns); (%o105) −%gamma Az 1-ben is konvergens: limit(-log(1-y)/y,y,0,plus); (%o102) 1 (2/1) integrate((-log(x))^(-log(x)),x,0,1); (%o149) 011

−log

x

log

x

dx
changevar(%,log(x)=-y,y,x); (%o150) 0infyy*%e−ydy limit(y^y/exp(y),y,inf); (%o151) inf (2/2) integrate(x^log(x),x,0,1); defint: integral is divergent. −− an error. To debug this try: debugmode(true); 'integrate(x^log(x),x,0,1); (%o156) 01xlog

x

dx
changevar(%,log(x)=-y,y,x); (%o157) 0inf%ey2−ydy limit(exp(y^2-y),y,inf); (%o158) inf (2/3) limit((sqrt(x+1)-sqrt(x))/x^(-1/2),x,inf); (%o160) 12 (3/1) limit(sin(sin(sin(x)))/x,x,0); (%o136) 1 (3/2) limit((x+1)/(x^2*sqrt(x^2-1))/x^(-2),x,inf); (%o141) 1 (3/3) limit(x^x/(x+1)!,x,inf); (%o143) inf (4/1) limit((2^x+3^x)^(1/x),x,inf); (%o144) 3 limit((x-log(x))*sqrt(x)/x^(3/2),x,inf); (%o145) 1 (4/2) limit(-log(cos(x)),x,%pi/2); (%o1) infinity (4/3) limit(sqrt(sin(x)),x,%pi/2); (%o161) 1 subst(y+%pi/2,x,1/cos(x)); (%o163) −1sin

y

limit(sin(y)/y,y,0); (%o164) 1 (4/3) integrate(1/(%pi/2-asin(x))^(3/2),x,0,1); (%o165) 011

%pi2−asin

x

32
dx
changevar(%,x=sin(y),y,x); (%o166) −232*0%pi2cos

y

%pi−2*asin

sin

y

*

2*asin

sin

y

−%pi

dy
trigsimp(%); (%o167) −232*0%pi2cos

y

%pi−2*asin

sin

y

*

2*asin

sin

y

−%pi

dy
integrate(cos(y)/(%pi/2-y)^(3/2),y,0,%pi/2); (%o171) 0%pi2cos

y

%pi2−y

32
dy
changevar(%,y-%pi/2=-x,x,y); (%o172) 0%pi2cos

2*x−%pi2

x32
dx
trigsimp(%); (%o173) 0%pi2cos

2*x−%pi2

x32
dx
cos(x-%pi/2); (%o174) sin

x

limit(sin(x)/x,x,0); (%o176) 1 (5/1) >1/x (5/2) >1/x Feladat (1) limit(sin(x)/x,x,0); (%o126) 1 (2) limit((1-cos(x))/x^2,x,0); (%o127) 12 (3) assume(c>0) limit((1+x^c)/x^c,x,inf); (%o129) 1 (4) assume(d<0); (%o133) [d<0] limit(((1-y)^d-1)/(1-y)^d,y,1,plus); (%o135) 1 Feladat (1) taylor(exp(-x^2),x,0,30); (%o114)/T/ 1−x2+x42−x66+x824−x10120+x12720−x145040+x1640320−x18362880+x203628800−x2239916800+x24479001600−x266227020800+x2887178291200−x301307674368000+... integrate(%,x,0,1); (%o115) 1302860390186918317445343996591875 float(%), numer; (%o116) 0.7468241328124257 (2) taylor(sin(x),x,0,30); (%o117)/T/ x−x36+x5120−x75040+x9362880−x1139916800+x136227020800−x151307674368000+x17355687428096000−x19121645100408832000+x2151090942171709440000−x2325852016738884976640000+x2515511210043330985984000000−x2710888869450418352160768000000+x298841761993739701954543616000000+... integrate(%/x,x,-1,1); (%o118) 11058788066795997194326290472795270891358445121856498214178748229222400000000 float(%), numer; (%o119) 1.892166140734366 (3) taylor(cos(x^2),x,0,30); (%o120)/T/ 1−x42+x824−x12720+x1640320−x203628800+x24479001600−x2887178291200+... integrate(%,x,0,1); (%o121) 9718863640675011074472439040000 float(%), numer; (%o122) 0.9045242379002706 Feladat Feladat Parciális integrálásokkal In:(n!)^2*4^n/(2*n+1)!; (%o3) 4n*n!2

2*n+1

!
Feladat A hasonló integrált csak koszinusszal Jelölje C[n]. Ekkor S[0]:1/2; C[0]:1/2; S[n]:n*S[n-1]/2+n*C[n-1]/2; C[n]:n*C[n-1]/2-n*S[n-1]/2; (%o12) 12(%o13) 12(%o14) Sn−1*n2+Cn−1*n2(%o15) Cn−1*n2−Sn−1*n2 Feladat G:F(x^4); (%o109) F

x4

diff(G,x); (%o110) dd*x*F

x4

4*x^3*exp(x^12)*sin(x^4); (%o111) 4*x3*%ex12*sin

x4

Feladat G:log(2*x)-log(x); (%o107) log

2*x

−log

x

diff(G,x); (%o108) 0 Nem függ x-től. Feladat solve(sin(t)=sqrt(t),t); (%o106) [sin

t

=t]
Feladat diff(sin(x)/(1+x),x); (%o112) cos

x

x+1
−sin

x

x+1

2
PK``lXñB–HmimetypePK``lXQdBV55 5format.txtPK``lX)ÈD£ü÷ü÷ ’content.xmlPK§·þ