mm-482 === Subject: Re: calc 1 and 2your comments did not strike me that way AT ALL. Plus, my hat's off to anyone who usee the Scylla of X and the Charybdis of Y as a metaphor --- thought I was the last one. :-)I'm truly looking for (dare I say) ideas outside the box, because what I've been doing in the past hasn't worked well for me or the students (not to mention their Calc II instructors). I like your idea of thinking of the course, partly, as being about where to find things.I have this pesky idea that if you get a passing grade in Calc I you ought to be able actually to do some calculus. If I could only give that up I'd be so much happier. :-) === Subject: Re: calc 1 and 2>Anyway, my guess is that you will compress rather than omit. Probably>everything that (the faculty thinks) can be omitted already has been.Maybe I'm not good enough, but I can't see how to compress any further. I honestly can't. Students in past summers have pretty generally done far less homework than they needed to; I fear if I compress further that the proportion will drop even lower (if possible).I'm thinking of a really radical change -- for instance, demanding that they study the textbook on their own, with _no_ lectures, and use class time to work the assigned problems. I'd have to make those problems a major component of their grade, of course.Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.comAn expense does not have to be required to be considered necessary. -- IRS Form 1040 line 23 instructions === Subject: Re: calc 1 and 2>Anyway, my guess is that you will compress rather than omit. Probably>everything that (the faculty thinks) can be omitted already has been.> Maybe I'm not good enough, but I can't see how to compress any > further. I honestly can't. Students in past summers have pretty > generally done far less homework than they needed to; I fear if I > compress further that the proportion will drop even lower (if > possible).> I'm thinking of a really radical change -- for instance, demanding > that they study the textbook on their own, with _no_ lectures, and > use class time to work the assigned problems. I'd have to make those > problems a major component of their grade, of course.I'm not sure I would recommend _that_. Unless you are fortunate enoughto have a grader or two, graded homework is not an option and quizzesare pretty much out of the question also.There is no good way to handle exams. If you're not careful an exam caneat up a day ( = a week in regular time).If your classes meet twice a day you can cover new material and make aassignment in the PM. That requires that the students know they aresupposed to work on the assignments in the interim.A major problem is that texts seem to written with 50 minute classes inmind and you will either find yourself covering a partial sections orfalling behind. There is _no_ way you can arrange for the students to have time forthings to sink in.I wouldn't feel bad leaving out graph sketching and spending minimaltime on max/min problems.There isn't a _good_ solution. Good luck.Paul SperryColumbia, SC (USA) === Subject: Re: calc 1 and 2Stan Brown said:<To be serious and consistant with the quote shown above, studying calculus 1 in>a 4 or 6 week summer course is a crazy thing to do. Rcall an earlier comment>(from me) about needing time to learn, and that this course requires maybe 3 or>4 months of student development time. Yes, I heard you the first time. I didn't reply directly to that point because I agree wholeheartedly and I don't expect anyone disagrees. A very unusual student might be able to do it faster and do it well (as I did, the summer before my last year of high school), but I think it's unreasonable to expect that of the great majority.Nevertheless, the college teaches this course in the summer, and students register for it, expecting that they'll then take Calc II in the fall. I have to do the best I can for them and for the College. === Subject: Re: calc 1 and 2>To be bluntly honest about a student trying to study this in a summer session,>the department or teacher should maybe orient to students in a properly refined>manner equivalent to something like, If you want to learn effectively from>this 6 week summer course, you better work like all hell is breaking loose,>because for you students, it is! I actually told them that last summer, in more or less those words. (I believe I used the phrase bat out of hell.) They looked suitably grave, but there was no apparent effect on the amount of work they did as compared to the previous summer's students.Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.comAn expense does not have to be required to be considered necessary. -- IRS Form 1040 line 23 instructions === Subject: Re: calc 1 and 2disclaimer-- I am not a teacher so take this FWIW> Stan Brown said:> < that they study the textbook on their own, with _no_ lectures, and> use class time to work the assigned problems. I'd have to make those> problems a major component of their grade, of course.> To be serious and consistant with the quote shown above, studying calculus1 in> a 4 or 6 week summer course is a crazy thing to do. Rcall an earliercomment> (from me) about needing time to learn, and that this course requires maybe3 or> 4 months of student development time.> To be bluntly honest about a student trying to study this in a summersession,> the department or teacher should maybe orient to students in a properlyrefined> manner equivalent to something like, If you want to learn effectivelyfrom> this 6 week summer course, you better work like all hell is breaking loose,> because for you students, it is!True also, of course.> Another idea to try is to require a summer-course qualification aptitudetest> for calculus 1. Such test could be administered before the summer course> begins. Students who do not pass it would not be permitted to attend the> summer calculus course.What would such a test consist of, if not calculus itself? Precalc stuff?Such 'tests' already exist in the form of whatever prerequisites alreadyexist for a 'normal' calcI offering. Now if this were some sort of 'honers'class, then you may be on to something, but it's not. It's ordinary calcI.> Calculus 1 for me was one of the damn-hardest courses I ever took. Thiswas in> a regular semester length term. I did not learn it adequately andrestudied it> on my own for 3 months before continuing.For me, cal I and III were relative cake walks, with cal II being the mostdemanding. Blood and guts of integration, and all that jazz. Anyone'smileage may vary. The point is, it's not really fair to the student _or_the instructor to have to take/teach any one of these courses in such ashort time, and to think the OP was considering taking both cal I and IIover the summer (he said the classes are 4 weeks, which I assume are percourse, i.e. 8 weeks for both). Whew.It's not so much the difficulty as it is the *amount* of material. LikeStan alluded to, corners need be cut somewhere. I remember when I took calIover a full fall semsester we had so many snow days that corners had to becut. The instructor cut the epsilon-delta stuff. He basically pointed usto the text and suggested we review it at will (he at least let us know itexisted), and let us know that it isn't really used much in practice exceptto prove other theorems and what not, then moved on to other stuff that wewould most likely see more of in the future. he spent about a minute and ahalf on it in the classroom. In retrospect, it seems that was as good acorner to cut as any. I did try to learn epsilon-delta proofs on my own,with limited success, but it really didn't matter because I really didn'tsee it much after that.One or two such cut corners, to me at least, seem manageable. Teaching ortrying to learn all of cal I and II over an 8 week period seems insane. Mypoint is everyone seems to _know its insane yet its still apparently beingdone, which says something. === Subject: Re: calc 1 and 2>Therefor you can cover the same material, right? Actually, _you_>probably can. Whether the students can is another question.Funny -- that sounds almost exactly like what the associate dean said to me.I give him marks for recognizing the absurdity. Not all administrators do.Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.comAn expense does not have to be required to be considered necessary. -- IRS Form 1040 line 23 instructions === Subject: Sintetic Geometry Problem by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i3H21t508342;Could anyone help me with this one?Let ABC be any triangle. Find a point D. Such that angle DAB = angleDBC = angle DCA. I would apreciate any help. === Subject: Re: number of indep. soloutions to diffyqs? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i3H21n308105;>I've taken some classes in solving differential equations, and Iunderstand>all the different techniques, and everything, but am confused as tohow we>can be assured that our techniques yield *all* possible solutions.>Especially when we do things like, if x1 is a solution, put x2(x) =>x1(x)*f(x) into the differential equation and obtain anothersolution. Is>there a general way to tell how many linearly independent solutions a>differential equation has, (especially regardless of homogeneity and>linearity)? I understand why these methods work, of course, butcan't see>exactly how we can be sure we are obtaining all the solutions. I cansee>intuitively a little why it should be the case (especially when using>eigenvalues of a matrix in the matrix form of a diffyq x'=Ax, butwould like>something more rigorous).>Jeremy The question of how many LINEARLY INDEPENDENT solutions a d.e. hasonly makes sense if the d.e. is linear itself. In that case, mostelementary d.e. texts have the proof that 1) The set of all solutions to a linear, homogeneous d.e. form avector space. 2) The dimension of that space (the number of linearly independentsolutions) is the same as the order of the differential equation. Roughly, the proof goes like this: One shows that, for a thirdorder differential equation (for example), the existence anduniqueness theorems show that there exist unique y1, y2, y3satisfying the differentential equation AND 1) y1(0)= 1, y1'(0)= 0, y1(0)= 0 2) y2(0)= 0, y2'(0)= 1, y2(0)= 0 3) y3(0)= 0, y3'(0)= 0, y3(0)= 1 (These are called the fundamental solutions at x= 0.) It is then easy to show that these functions form a basis for thevector space of all solutions. The extension to any order linear, homogenous d.e. should beobvious. The set of solutions to a non-homogeneous linear d.e. do not form avector space, but one can show that, if u(x) is any one solution tothe entire equation, then any solution can be written y= yh(x)+ u(x)where yh(x) is a solution to the associated homogeneous equation. === Subject: Re: Principle of Induction proof (LOGIC) - I'm a little fuzzyalt.math.undergrad:> In a book on logic I'm asked to do the following exercise:6.a> Show that if v1 and v2 are truth assignments which agree on> all the sentence symbols of the wff a, then v1`(a) = v2'(a).> To explain my notation, v is a truth function on a set of sentence> symbols S into {T, F}. Its extension denoted as v`, has a domain S`> which can be thought of as the set of all wffs that can be built up from> the sentence symbols in S, and range {T, F}.> So now my question: unless I'm misunderstanding things, the hypothesis> is that v1 = v2 ?The hypothesis is that if T is the set of sentence symbolsoccurring in a, then v1|T = v2|T; this doesn't require v1 toagree with v2 on sentence symbols not occurring in a.> Which to me, leads me to think v1`(a) = v2`(a) is a> trivial consequence ... yet I'm asked to prove this using the induction> principle, which doesn't seem to me to be the simplest method of proof?It's just about the only way.> But... assuming this is what's expected of me I'm not sure how I should> go about the inductive proof. My understanding of the induction> principle discussed in this book is a little fuzzy. I think I have a decent> elementary grasp of numerical induction, I can inductively prove basic> properties of sequences and functions by algebraically manipulating an> equation (i.e. 1 + 2 + ... n = n(n+1)/2).> But in this case, the _Induction Principle_ is described as:If S is a set of wffs containing all the sentence symbols and> closed under all five formula-building operations, then S is> the set of all wffs.> Which seems a bit different than the induction method I've been> exposed to previously, Same basic idea. The one that you've previously beenexposed to boils down to this: if S is a set of positiveintegers containing 1 and closed under the successoroperation (i.e., adding 1), then S = Z+. Typically S is theset of n in Z+ for which some assertion is true, like thesummation formula in your example; the 'basis step' isverifying that 1 is in S; and the 'induction step' isverifying that S is closed under successor.Here you have to do the same sort of thing: the 'basis step'is showing that all sentence symbols are in S, and the'induction step' is showing that S is closed under the fiveformula-building operations. I don't know what yourparticular five operations are, but one of them may beconjunction: if e and f are wffs, so is e & f; if so, you'dhave to show that if e and f are in S, then so is e & f.Let W be the set of all wffs containing only sentencesymbols in T (defined as before), and let S be the set ofall wffs b in W such that v1`(b) = v2`(b); you'd like toshow that a is in S, and this would certainly beaccomplished if you could show that S is all of W. The'basis step' is trivial: you're given it as a hypothesis (v1and v2 agree on T). If conjunction is one of youroperations, you have to prove that if v1`(e) = v2`(e) andv1`(f) = v2`(f), then v1`(e & f) = v2`(e & f). Then youhave to do the same sort of thing with of the otheroperations.The basic idea is that just as the positive integers arebuilt up from 1 by repeated addition of 1, the wffs over agiven set of sentence symbols are built up from thosesymbols by repeated application of a limited number offormula-building rules. If 1 has a certain property, andadding 1 preserves that property, then every positiveinteger must have it: that's the induction with which you'refamiliar. If all sentence symbols have a certain property,and the formula-building rules preserve that property, thenevery wff built from those symbols has the property: that'sthe induction that you're using here.[...]Brian === Subject: Re: Integral =< 118/3> Suppose that f(x)=ax^2+bx+c , a,b,c in R, is such that> (1) |f(x)| =< 1 for all x in [0,1] .> It's true that > Integral_{t=1 to t=3}|f(t)| dt =< 118/3 ?> Yes! However, you must justify that (1) implies (2) |f(t)| =< 8t^2 -8t+1 where t is outside from [0,1].> It should be clear, by drawing a sketch if necessary, [...] === Subject: Re: Integral =< 118/3>Suppose that f(x)=ax^2+bx+c , a,b,c in R, is such that> (1) |f(x)| =< 1 for all x in [0,1] .>It's true that > Integral_{t=1 to t=3}|f(t)| dt =< 118/3 ?> Yes! > However, you must justify that (1) implies> (2) |f(t)| =< 8t^2 -8t+1 where t is outside from [0,1].> It should be clear, by drawing a sketch if necessary, [...]Did you try to sketch the situation? Your line or parabola must lie between the upper and lower horzontal line segments and be as steeply rising as possible to the right of them, from (1,1) upwards to the highest possible (3,y).A little experimentation should indicate why you should chose the parabola passing through the ponts marked with an 'x'. A formal argument would be more difficult, and no more enlightening. | |y = 1 x-----------x | | |-----------0-----------1--------------- | | |y = -1 |-----x------ | |Among all lines and parabolas satisfying the constraints, none can be higher at any point of the interval of integration, [1,3], than the parabola passing through points (0,1), (1/2,-1) and (1,1), which is f(x) = 8*x^2 - 8*x + 1.One could get the same result, but not a better one, using f(x) = -8*x^2 + 8*x -1, with the vertex of the parabola at its highest point. === Subject: AskMars.com - get answers to your questions - Experts get paid Folks,Do you have a question? Get answers now!Are you an expert on an specific topic?50% of the value of the question is for you!http://www.askMars.com - Person to Person Knowledge SharingCurrent questions: === Subject: i need an answerValue: $ 40Question: pga golfer wears a lime green sweatshirt on the final day of golffirst and last name === Subject: the windows task maneger boxValue: $ 1Question: now my computer had a 80 gig hard drive but now it says i haveless than 8 g's left and my system needs at least 15% free space to propelydefrage my computer.i know where alot of the space has went but not 72 gigsworth of stuff.so im trying to clean out the things that are not needed andi hit a snag...so i have two problems and two questions.one what wouldhappen if i went into the windows task maneger and clicked on everythingthere and hit end task??would that shut my computer down or delete stuffthat i need?also when i go into add or remove programs and try to delete oldprograms or stuff i dont want on my computer it say's (could not openInstall.log file)without the parenthasees i know that last word is spelledwrong but it's the best i could do.will somebody please help me my backsagainst the wall here. === Subject: anwsers for my final exam.Value: $ 1Question: Where can I find the site that has the anwsers to thAmerican/Arizona Goverment Final Exam? I realy need this to graduate. === Subject: integral by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i3HChL808003;integeral form (-1 to 1) for ( 1/(1+x^3+square root for (1+x^6))dx) === Subject: Re: integral> integeral form (-1 to 1) for ( 1/(1+x^3+square root for (1+x^6))dx)Rationalize the demoninator and get on with it.1/(1 + x^3 + sqr(1 + x^6))= (1 + x^3 - sqr(1 + x^6) / (1 + 2x^3 + x^6 - (1 + x^6))= (1 + x^3 - sqr(1 + x^6) / 2x^3= (1/2)(1 + 1/x^3 - sqr(1 + 1/x^6))= hm... === Subject: Exactly Similar!?I am not a mathematician, but I know that the term (or combination ofwords); Exactly Similar, is a valid description of an object/condition inMathematics... I can find many uses of this term but a respectable sitewith a clear definition eludes me! === Subject: Re: Exactly Similar!?> I am not a mathematician, but I know that the term (or combination of> words); Exactly Similar, is a valid description of an object/condition> in Mathematics...I don't say that it's not valid, but I cannot recall that phrase havingbeen used in mathematics. In all mathematical contexts known to me,similar _by itself_ is exact. Thus, in the phrase exactly similar,exactly would be superfluous.> I can find many uses of this termHmm. Please mention some mathematical uses in which exactly similar meanssomething different from similar.> but a respectable site with a clear definition eludes me!Maybe there's a reason for that. ;-) === Subject: Re: Exactly Similar!?Here are some uses of this term:http://nrich.maths.org/public/viewer.php?obj_id=1326& refpage=monthindex.php&part=index&nomenu=1&PHPSESSID= 010f626cebc67c30c2be882e19bd89ceExactly similar arguments illustrated by similar pictures show that at alowest point the ground is flat. - Dr Tom K.9arner (Mathematics, Univ.Cambridge)http://ocw.mit.edu/NR/rdonlyres/ Civil-and-Environmental-Engineering/1- 00Introduction-to-Computers-and-Engineering-Problem-SolvingSpri ng2002/97834FAF-78A7-447C-9877-49125B23E2C7/0/tutorial4. pdfThis is exactly similar to the conditional statements in a while or a forloop which are used to terminate the iteration. (Page 4)I have many more, papers from various universities in US and UK...> I am not a mathematician, but I know that the term (or combination of> words); Exactly Similar, is a valid description of an object/condition> in Mathematics...> I don't say that it's not valid, but I cannot recall that phrase having> been used in mathematics. In all mathematical contexts known to me,similar _by itself_ is exact. Thus, in the phrase exactly similar,exactly would be superfluous.> I can find many uses of this term> Hmm. Please mention some mathematical uses in which exactly similarmeans> something different from similar.> but a respectable site with a clear definition eludes me!> Maybe there's a reason for that. ;-) === Subject: Re: Exactly Similar!?Exactly similar arguments illustrated by similar pictures show that at a>lowest point the ground is flat. - Dr Tom K.9arner (Mathematics, Univ.>Cambridge)This is exactly similar to the conditional statements in a while or a for>loop which are used to terminate the iteration. (Page 4)Those are not mathematics, but English. And (in my opinion) they're both wrong, just like writing relatively unique.Similar means nearly the same, but not completely the same. It's an oxymoron to call two things exactly similar.Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.comAn expense does not have to be required to be considered necessary. -- IRS Form 1040 line 23 instructions === Subject: Re: Exactly Similar!?Similar means nearly the same, but not completely the same. It's> an oxymoron to call two things exactly similar.Just to nitpick, not necessarily.Two things can be similar and yet the same in all waysthat matter, e.g. two triangles with vertices (0,0),(0,1),(1,0) and(1,0),(1,1),(2,0) respectively. They are essentially the same (congruent,or 'exactly' the same, if you will) yet they are also similar (aren'tthey??) but in most cases even when they are both 'exact' and 'similar' itis at best superfluous to say both, and at worse a contradiction (oxymoron).In most cases where it may be technically OK to use both, it is probably notthe best choice of words, and, in most cases its probably not the case thatthey are exact to begin with. === Subject: Re: Exactly Similar!?> Here are some uses of this term:> http://nrich.maths.org/public/viewer.php?obj_id=1326&refpage= monthindex.p> hp&part=index&nomenu=1&PHPSESSID= 010f626cebc67c30c2be882e19bd89ceExactly similar arguments illustrated by similar pictures show that at a> lowest point the ground is flat. - Dr Tom K.9arner (Mathematics, Univ.> Cambridge)> http://ocw.mit.edu/NR/rdonlyres/ Civil-and-Environmental-Engineering/1-00I> ntroduction-to-Computers-and-Engineering-Problem-SolvingSpring2 002/97834F> AF-78A7-447C-9877-49125B23E2C7/0/tutorial4.pdfThis is exactly similar to the conditional statements in a while or a> for loop which are used to terminate the iteration. (Page 4)As someone (Peter Webb, perhaps) said to you when you asked this samequestion in another newgroup, the quotations above show merely that theauthor was being sloppy!> I have many more, papers from various universities in US and UK...Maybe the explanation is simply that there are many sloppy people, even inacademia.>stake!>I am not a mathematician, but I know that the term (or combination of>words); Exactly Similar, is a valid description of an>object/condition in Mathematics...> I don't say that it's not valid, but I cannot recall that phrase having> been used in mathematics. In all mathematical contexts known to me,similar _by itself_ is exact. Thus, in the phrase exactly similar,exactly would be superfluous.>I can find many uses of this term> Hmm. Please mention some mathematical uses in which exactly similar> means> something different from similar.>but a respectable site with a clear definition eludes me!> Maybe there's a reason for that. ;-) === Subject: Re: Exactly Similar!?This is just sloppy argument - see my reply in other group!> Here are some uses of this term:http://nrich.maths.org/public/viewer.php?obj_id=1326& refpage=monthindex.p> hp&part=index&nomenu=1&PHPSESSID= 010f626cebc67c30c2be882e19bd89ceExactly similar arguments illustrated by similar pictures show that ata> lowest point the ground is flat. http://ocw.mit.edu/NR/rdonlyres/ Civil-and-Environmental-Engineering/1- 00Introduction-to-Computers-and-Engineering-Problem-SolvingSpri ng2002/97834F> AF-78A7-447C-9877-49125B23E2C7/0/tutorial4.pdfThis is exactly similar to the conditional statements in a while or a> for loop which are used to terminate the iteration. (Page 4)> As someone (Peter Webb, perhaps) said to you when you asked this same> question in another newgroup, the quotations above show merely that the> author was being sloppy!> I have many more, papers from various universities in US and UK...> Maybe the explanation is simply that there are many sloppy people, even >I am not a mathematician, but I know that the term (or combinationof>words); Exactly Similar, is a valid description of an>object/condition in Mathematics...>I don't say that it's not valid, but I cannot recall that phrasehaving>been used in mathematics. In all mathematical contexts known to me,similar _by itself_ is exact. Thus, in the phrase exactly similar,exactly would be superfluous.>I can find many uses of this term>Hmm. Please mention some mathematical uses in which exactly similar> means>something different from similar.>but a respectable site with a clear definition eludes me!>Maybe there's a reason for that. ;-) === Subject: Just verifying a study problemI know this is probobly very basis but the book does not have the answerfor me to go back and check.The problem:For what value of 'r' will the 3 column vectors be linearly dependent:1 -2 22 -4 r-3 6 1Perhaps it's the wording of the question (ie, a trick one) which makesme pay more attention to it:After row-reducing, I get:1 -2 20 0 -2+r0 0 7So I'm saying no values of 'r' will make these dependent simply due tothe 3rd row (0 0 7) implying an inconsistent solution. Now if the 3rdrow was all zeros (or some valid combination between the columns), thenI'd conclude 'r' to be 2. If I am wrong on this, please blast me!!! === Subject: Re: Just verifying a study problem> I know this is probobly very basis but the book does not have the answer> for me to go back and check.> The problem:> For what value of 'r' will the 3 column vectors be linearly dependent:> 1 -2 2> 2 -4 r> -3 6 1Notice that (column 2) = -2 * (column 1). So just from that you knowthat it is impossible for the three columns to be linearly*in*dependent and therefore for *any* value of r, the columns will belinearly *de*pendent.> After row-reducing, I get:> 1 -2 2> 0 0 -2+r> 0 0 7> So I'm saying no values of 'r' will make these dependent simply due to> the 3rd row (0 0 7) implying an inconsistent solution. Now if the 3rd> row was all zeros (or some valid combination between the columns), then> I'd conclude 'r' to be 2. If I am wrong on this, please blast me!!!You're wrong.Remember, if the determinant of a matrix is zero, its columns arelinearly *dependent*. Also remember that row reduction leaves thedeterminant unchanged. It is clear from inspection that thedeterminant of your row-reduced matrix is zero for any r. Thereforethe determinant of the original matrix is zero for any r. Thereforethe columns of the original matrix are linearly dependant for any r. === Subject: Re: finding x,y intercepts by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i3HKZ4L28041;Having trouble finding intercepts of -3y+4x=7Help? === Subject: Re: finding x,y intercepts> Having trouble finding intercepts of -3y+4x=7The x-intercept, by definition, is the place where the lineintercepts or crosses the x-axis. In other words, it is the point(a, 0), where a is the value that makes y be 0.Similarly, the y-intercept is the place where the line interceptds they-axis, i.e. the point (0, b), where b is the value that makes x be 0.So to find the y-intercept, substitute the point (0, b) into the givenequation and solve to get the value of b. === Subject: integration by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i3HKZ3828025;I have that the integration of ( 3x^2 + 1 ) / ( ( 3x^2 - 1 )^2 ) isequal to - x / ( 3x^2 - 1 ). Can't understand how to get this answer, === Subject: Re: integration> I have that the integration of ( 3x^2 + 1 ) / ( ( 3x^2 - 1 )^2 ) is> equal to - x / ( 3x^2 - 1 ). Can't understand how to get this answer,Use Partial Fractions.(3x^2+1)/(3x^2-1)^2=(Ax+B)/(3x^2-1)+(Cx+D)/(3x^2-1)^ 23x^2+1=(Ax+B)(3x^2-1)+Cx+D3x^2+1=3Ax^3-Ax+3Bx^2-B+Cx+DNow you need a system of equations by equating coefficients3A=03B=3-A+C=0-B+D=1Solving, we get A=0, B=1, C=0 D=2So our function becomes1/(3x^2-1)+2/(3x^2-1)^2Now just use an integral table.David Moran