UNIVERSITY OF CALIFORNIADepartment of Electrical Engineeringand Computer SciencesComputer Science DivisionCS 282 Prof. R. FatemanFall 2002Assignment 4: Simplification, IntegrationDue: Wed, 27 March, 20021. Experiment with the integration programs in Maple, Mathematica, and/or Macsyma.See if you can come up with examples of indefinite integrals that exist in closed form in termsof elementary functions but which they do not find. One way of doing this is to start withexpressions, and differentiate (and rearrange). One heuristic that sometimes causes problemsis integrating a sum of terms by integrating term-by-term. What if the individual terms arenot integrable, but the sum is?Can you outline some region(s) of “expression space” that characterize your discoveries?2. Definite improper integrals (limits including ∞, or integrals with singularities at theendpoints or in the middle), cause problems sometimes. Consider the integrand 1/(x−a)2andvarious endpoints. What should the answer be? Experiment w ith the integration programsin Maple, Mathematica, and/or Macsyma. See if you can come up with examples of definite(symbolic) integrals with parameters that they do not get correct.3. All univariate real polynomials can be factored “numerically” into (at worst) quadraticfactors over the reals, and into (at worst) linear factors over the complex numbers. Thistrivializes rational function integration. Attack and/or defend this p osition.4. This problem concerns iteration in a power series domain. The short Macsyma programbelow uses “Picard’s method” (described in any elementary differential equations book) tosolve a first-order ordinary differential equation by integration, but in a power-series domain.Write a (short) program that will solve simple second-order ODEs using the same technique.(Solving simple ODEs would include, for example, finding x(t) whered2xdt2= f(x,dxdt, t)1Assignment 4: Simplification, Integration 2for f() is a polynomial in three variables.) Picard, under some restrictions, solves the equationdxdt= f(x, t)where x(0) = a0 as a series to degree n in t. For example, you could try picard(x,x,t,1,5);picard(f,x,t,a0,n):=block([s:a0,deg:0],while deg < n do(deg:deg+1, s:ratdisrep(s),s:integrate(taylor(subst(x=s,f),t,0,deg),t)+a0),return(taylor(s,t,0,n)));To better understand picard, you can use debugging information from setcheck:[s,deg];and trace(integrate,taylor); etc. There is an analogy to p-adic convergence in this busi-ness, but you need not discuss it.Show that your program can solve y00+ y = 0, y(0) = 0, y0(0) = 1.Feel free to use Mathematica or Maple or Mupad instead of Macsyma. If you need helpin deciphering the program, see me.5. Consider the algorithm discussed by Moses for testing for zero equivalence of an expressionin a class which differs from Brown’s REX expressions in that it involves no i, only a singlevariable x but allows the function log(|x|). The log as well as the exponential can be nestedto any depth. This algorithm (also due to D. Richardson) works by reducing the decisionprocess to one requiring the solution to “the constant problem” of determining whether anexpression consisting entirely of constants is zero or not.(a) What limitations are relevant on this algorithm. Give examples where these limitationscome into play. (Hint: Is this expression a constant: arctan(x) + arctan(1/x)?)(b) Describe alternative algorithms or heuristics for solving this constant problem.(c) What properties of the derivative are used by Richardson?(d) Consider the extension to Richardson’s results suggested by Moses so that the class caninclude functions spe cifically defined by a differential equation of the form y0= a(x)y + b(x).Describe the algorithmic details for this, and if you can, write a program for it.For example, the equation y0= (2/√π)e−x2has as its solution a function y = erf(x)that is an integral not expressible in terms of elementary functions. Can you show thaterf(x) + erf(−x) is