A.15APPENDIX B Mathematics ReviewThese appendices in mathematics are intended as a brief review of operations andmethods.Earlyinthiscourse,youshouldbetotallyfamiliarwithbasicalgebraictechniques,analyticgeometry,andtrigonometry.Theappendicesondifferentialandintegralcalculusaremoredetailedandareintendedforthosestudentswhohave difculty applying calculus concepts to physical situations.SCIENTIFIC NOTATIONManyquantitiesthatscientistsdealwithoftenhaveverylargeorverysmall values.Forexample,thespeedoflightisabout300000000m/s,andthe inkrequiredtomakethedotoverani inthistextbookhasamassofabout 0.000 000 001 kg. Obviously, it is very cumbersome to read, write, and keep trackof numbers such as these. We avoid this problem by using a method dealing withpowers of the number 10:andsoon.Thenumberofzeroscorrespondstothepowertowhich10israised,called the exponent of 10. For example, the speed of light, 300 000 000 m/s, canbe expressed as 3 108m/s.In this method, some representative numbers smaller than unity are105110 10 10 10 10 0.000 01104110 10 10 10 0.000 1103110 10 10 0.001 102110 10 0.01 101110 0.1 105 10 10 10 10 10 100 000104 10 10 10 10 10 000 103 10 10 10 1000 102 10 10 100 101 10 100 1 B.1A.16 APPE NDI X BIn these cases, the number of places the decimal point is to the left of the digit 1equals the value of the (negative) exponent. Numbers expressed as some power of10 multiplied by another number between 1 and 10 are said to be in scientic no-tation. Forexample,thescienticnotationfor5943000000is5.943 109andthat for 0.000 083 2 is 8.32 105.Whennumbersexpressedinscienticnotationarebeingmultiplied,thefol-lowing general rule is very useful:(B.1)wheren andm canbeany numbers(notnecessarilyintegers).Forexample,Therulealsoappliesifoneoftheexponentsisnegative:When dividing numbers expressed in scientic notation, note that(B.2)EXERCISESWith help from the above rules, verify the answers to the following:1. 86 400 8.64 1042. 9 816 762.5 9.816 762 5 1063. 0.000 000 039 8 3.98 1084. (4 108)(9 109) 3.6 10185. (3 107)(6 1012) 1.8 1046.7.ALGEBRASome Basic RulesWhenalgebraicoperationsareperformed,thelawsofarithmeticapply.Symbolssuchasx,y,andz areusuallyusedtorepresentquantitiesthatarenotspecied,what are called the unknowns.First, consider the equationIf we wish to solve for x, we can divide (or multiply) each side of the equation bythe same factor without destroying the equality. In this case, if we divide both sidesby 8, we have x 4 8x83288x 32B.2(3 106)(8 102)(2 1017)(6 105) 2 101875 10115 103 1.5 10710n10m 10n 10m 10nm103 108 105.102 105 107.10n 10m 10nmB.2 Algebra A.17Next consider the equationInthistypeofexpression,wecanaddorsubtractthesamequantityfromeachside. If we subtract 2 from each side, we getIn general, ifthen Now consider the equationIf we multiply each side by 5, we are left with x on the left by itself and 45 on theright:In all cases, whatever operation is performed on the left side of the equality must also be per-formed on the right side.Thefollowingrulesformultiplying,dividing,adding,andsubtractingfrac-tions should be recalled, where a, b, and c are three numbers: x 45 x5 (5) 9 5x5 9x b a. x a b, x 6 x 2 2 8 2x 2 8Rule ExampleMultiplyingDividingAdding2345(2)(5) (4)(3)(3)(5) 215abcdad bcbd2/34/5(2)(5)(4)(3)1012(a/b)(c/d)adbc23 45 815 ab cd acbdEXERCISESIn the following exercises, solve for x:Answers1.2.3.4.PowersWhen powers of a given quantity x are multiplied, the following rule applies:(B.3) xnxm xnmx 11752x 634x 8x 7a bax 5 bx 2x 6 3x 5 13x 1 aaa 11 xA.18 APPE NDI X BFor example, When dividing the powers of a given quantity, the rule is(B.4)For example, A power that is a fraction, such ascorresponds to a root as follows:(B.5)For example,(A scientic calculator is useful for such calcula-tions.)Finally, any quantity xnraised to the mth power is(B.6)Table B.1 summarizes the rules of exponents.EXERCISESVerify the following:1.2.3.4. (Use your calculator.)5. (Use your calculator.)6.FactoringSome useful formulas for factoring an equation areQuadratic EquationsThe general form of a quadratic equation is(B.7)where x is the unknown quantity and a, b, and c are numerical factors referred toas coefcients of the equation. This equation has two roots, given by(B.8)Ifthe roots are real. b2 4ac,x b !b2 4ac2aax2 bx c 0a2 b2 (a b)(a b)differences of squaresa2 2ab b2 (a b)2 perfect square ax ay az a(x y x)common factor (x4)3 x12601/4 2.783 15851/3 1.709 975x10/x5 x15x5x8 x332 33 243(xn)m xnm41/3 !34 1.5874.x1/n !nx13 ,x8/x2 x82 x6.xnxm xnmx2x4 x24 x6.TABLEB.1Rules of Exponents(xn)m xnmx1/n !nxxn/xm xnmxnxm xnmx1 xx0 1B.2 Algebra A.19EXERCISESSolve the following quadratic equations:Answers1.2.3.Linear EquationsA linear equation has the general form(B.9)where m and b are constants. This equation is referred to as being linear becausethegraphofy versusx isastraightline,asshowninFigureB.1.Theconstantb,called the y-intercept, represents the value of y at which the straight line intersectsthey axis.Theconstantm isequaltotheslope ofthestraightlineandisalsoequaltothetangentoftheanglethatthelinemakeswiththex axis.Ifanytwopoints on the straight line are specied by the coordinates (x1, y1) and (x2, y2), asin Figure B.1, then the slope of the straight line can be expressed as(B.10)Notethatm andb canhaveeitherpositiveornegativevalues.Ifthestraightlinehasapositive slope,asinFigureB1.Ifthestraightlinehasanegative slope. In Figure B.1, both m and b are positive. Three other possible situa-tions are shown in Figure B.2.EXERCISES1. Draw graphs of the following straight lines:(a) (b) (c)2. Find the slopes of the straight lines described in Exercise 1.Answers (a) 5 (b)2 (c)3y 3x 6 y 2x 4 y 5x 3m 0,m 0,Slope y2 y1x2 x1yx tan y mx bx 1 !22/2 x 1 !22/2 2x2 4x 9 0x 12x 2 2x2 5x 2 0x 3 x 1 x2 2x 3 0EXAMPLE 1The equationhas the following roots corresponding to the two signs ofthe square-root term:where xrefers to the root corresponding to the positive sign and xrefers to the rootcorresponding to the negative sign.4 x 5 32 1 x 5 32 x 5 !52 (4)(1)(4)2(1)5 !925 32x2 5x 4 0y(x1, y1)(x2, y2)yx (0, b)(0, 0)xy(1)(2)(3)m > 0b < 0m < 0b > 0m < 0b < 0xFigure B.1Figure B.2A.20 APPE NDI X B3. Findtheslopesofthestraightlinesthatpassthroughthefollowingsetsofpoints:(a) (0, 4) and (4, 2), (b) (0, 0) and (2, 5), and (c) (5, 2) and (4, 2)Answers (a) 3/2 (b)5/2 (c)4/9Solving Simultaneous Linear EquationsConsidertheequationwhichhastwounknowns,x andy.Suchanequation does not have a unique solution. For example, note that (andare all solutions to this equation.Ifaproblemhastwounknowns,auniquesolutionispossibleonlyifwehavetwo equations.Ingeneral,ifaproblemhasn unknowns,itssolutionrequiresnequations.Inordertosolvetwosimultaneousequationsinvolvingtwounknowns,x and y, we solve one of the equations for x in terms of y and substitute this expres-sion into the other equation.(x 2, y 9/5) (x 5, y 0),x 0, y 3),3x 5y 15,EXAMPLE 2AlternateSolution Multiplyeachtermin(1)bythefactor 2 and add the result to (2):3 y x 2 x 1 12x 12 2x 2y 4 10x 2y 16 Solve the following two simultaneous equations:(1)(2)Solution From (2),Substitution of this into (1)gives1 x y 2 y 3 6y 18 5(y 2) y 8 x y 2.2x 2y 45x y 8Two linear equations containing two unknowns can also be solved by a graphi-cal method. If the straight lines corresponding to the two equations are plotted ina conventional coordinate system, the intersection of the two lines represents thesolution. For example, consider the two equationsTheseareplottedinFigureB.3.Theintersectionofthetwolineshasthecoordi-natesThis represents the solution to the equations. You should checkthis solution by the analytical technique discussed above.EXERCISESSolve the following pairs of simultaneous equations involving two unknowns:Answers1.x y 2x 5, y 3 x y 8x 5, y 3.x 2y 1 x y 2 54321x 2y = 11 2 3 4 5 6(5, 3)xx y = 2yFigure B.3B.3 Geometry A.212.3.LogarithmsSuppose that a quantity x is expressed as a power of some quantity a:(B.11)The number a is called the base number. The logarithm of x with respect to thebase a is equal to the exponent to which the base must be raised in order to satisfythe expression (B.12)Conversely, the antilogarithmof y is the number x:(B.13)In practice, the two bases most often used are base 10, called the common loga-rithmbase,andbase. . . ,calledEulersconstantorthenatural loga-rithm base. When common logarithms are used,(B.14)When natural logarithms are used,(B.15)Forexample,log1052 1.716,sothatantilog101.716 101.716 52.Likewise, lne52 3.951, so antilne3.951 e3.951 52.Ingeneral,notethatyoucanconvertbetweenbase10andbasee withtheequality(B.16)Finally, some useful properties of logarithms areGEOMETRYThe distance d between two points having coordinates (x1, y1) and (x2, y2) is(B.17) d !(x2 x1)2 (y2 y1)2B.3log(ab) log a log blog(a/b) log a log blog(an) n log aln e 1ln ea aln1a ln alne x (2.302 585) log10 xy lne x(or x e y)y log10 x(or x 10y)e 2.718x antiloga yy loga xx a y:x a y8x 4y 28x 2, y 3 6x 2y 6T 49 5aT 65, a 3.27 98 T 10aA.22 APPE NDI X BRadian measure: The arc length s of a circular arc (Fig. B.4) is proportionalto the radius r for a xed value of (in radians):(B.18)Table B.2 gives the areas and volumes for several geometric shapes used through-out this text:s r srTABLEB.2 Useful Information for GeometryShape Area or Volume Shape Area or VolumeRectanglewrCircleTrianglehSphererCylinderRectangular boxrVolume = r2Surface area = 4r2Area =2(h + w + hw)Volume = whwhArea = r2(Circumference = 2r)Area = wb Area = bh12Volume =4r33Lateral surfacearea = 2r The equation of a straight line (Fig. B.5) is(B.19)where b is the y-intercept and m is the slope of the line.The equation of a circle of radius R centered at the origin is(B.20)The equation of an ellipse having the origin at its center (Fig. B.6) is(B.21)where a is the length of the semi-major axis (the longer one) and b is the length ofthe semi-minor axis (the shorter one).x2a2y2b2 1x2 y2 R2y mx brsFigure B.4b0ym = slope = tanxFigure B.5y0baxFigure B.6B.4 Trigonometry A.23The equation of a parabola the vertex of which is at(Fig. B.7) is(B.22)The equation of a rectangular hyperbola (Fig. B.8) is(B.23)TRIGONOMETRYThatportionofmathematicsbasedonthespecialpropertiesoftherighttriangleiscalled trigonometry. By denition, a right triangle is one containing a 90 angle. Con-sider the right triangle shown in Figure B.9, where side a is opposite the angle , side bis adjacent to the angle , and side c is the hypotenuse of the triangle. The three basictrigonometric functions dened by such a triangle are the sine (sin), cosine (cos), andtangent (tan) functions. In terms of the angle , these functions are dened byThePythagoreantheoremprovidesthefollowingrelationshipbetweenthesides of a right triangle:(B.27)From the above denitions and the Pythagorean theorem, it follows thatThe cosecant, secant, and cotangent functions are dened byThe relationships below follow directly from the right triangle shown in Figure B.9:Some properties of trigonometric functions areThe following relationships apply to any triangle, as shown in Figure B.10: 180tan () tan cos () cos sin () sin cot tan(90 )cos sin(90 )sin cos(90 )csc 1sin sec 1cos cot 1tan tan sin cos sin2 cos2 1c2 a2 b2(B.24)(B.25)(B.26)sin side opposite hypotenuseaccos side adjacent to hypotenusebctan side opposite side adjacent to abB.4xy constanty ax2 by byb0xFigure B.70yxFigure B.8a = opposite sideb = adjacent sidec = hypotenuse90cab90Figure B.9A.24 APPE NDI X BLaw of cosinesLaw of sinesTable B.3 lists a number of useful trigonometric identities.asin bsin csin a2 b2 c2 2bc cos b2 a2 c2 2ac cos c2 a2 b2 2ab cos a = 2cb = 5Figure B.11TABLEB.3 Some Trigonometric Identitiescos(A B) cos A cos B sin A sin Bsin(A B) sin A cos B cos A sin Btan 2!1 cos 1 cos tan 22 tan 1 tan2 1 cos 2 sin2 2cos 2 cos2 sin2 cos2 212(1 cos ) sin 2 2 sin cos sin2 212(1 cos ) sec2 1 tan2 csc2 1 cot2 sin2 cos2 1EXAMPLE 3where tan1(0.400) is the notation for angle whose tangentis 0.400, sometimes written as arctan (0.400).ConsidertherighttriangleinFigureB.11,inwhich and c is unknown. From the Pythagorean theorem, wehaveTo nd the angle , note thatFrom a table of functions or from a calculator, we have21.8 tan1 (0.400) tan ab25 0.4005.39 c !29 c2 a2 b2 22 52 4 25 29b 5,a 2,EXERCISES1. InFigureB.12,identify(a)thesideopposite and(b)thesideadjacenttoand then nd (c) cos , (d) sin , and (e) tan .Answers (a) 3, (b) 3, (c) (d) and (e)2. Inacertainrighttriangle,thetwosidesthatareperpendiculartoeachotherare 5 m and 7 m long. What is the length of the third side?Answer 8.60 m4345 ,45 ,abcFigure B.10543Figure B.12B.6 Differential Calculus A.253. Arighttrianglehasahypotenuseoflength3 m,andoneofitsanglesis30.What is the length of (a) the side opposite the 30 angle and (b) the side adja-cent to the 30 angle?Answers (a) 1.5 m, (b) 2.60 mSERIES EXPANSIONSx in radiansForthe following approximations can be used1:DIFFERENTIAL CALCULUSInvariousbranchesofscience,itissometimesnecessarytousethebasictoolsofcalculus,inventedbyNewton,todescribephysicalphenomena.Theuseofcalcu-lus is fundamental in the treatment of various problems in Newtonian mechanics,electricity,andmagnetism.Inthissection,wesimplystatesomebasicpropertiesand rules of thumb that should be a useful review to the student.First,afunction mustbespeciedthatrelatesonevariabletoanother(suchas a coordinate as a function of time). Suppose one of the variables is called y (thedependentvariable),theotherx (theindependentvariable).Wemighthaveafunction relationship such asIf a, b, c, and d are specied constants, then y can be calculated for any value of x.Weusuallydealwithcontinuousfunctions,thatis,thoseforwhichy variessmoothly with x.y(x) ax3 bx2 cx dB.6ln(1 x) xtan x xex 1 xcos x 1(1 x)n 1 nxsin x xx V1,tan x x x332x515 x /2 cos x 1 x22!x44! sin x x x33!x55! ln(1 x) x 12x213x3 ex 1 x x22!x33! (1 x)n 1 nx n(n 1)2! x2 (a b)n ann1! an1b n(n 1)2! an2b2 B.51The approximations for the functions sin x, cos x, and tan x are for x 0.1 rad.A.26 APPE NDI X BThe derivative of y with respect to x is dened as the limit, as x approacheszero,oftheslopesofchordsdrawnbetweentwopointsonthey versusx curve.Mathematically, we write this denition as(B.28)wherey andx aredenedasand(Fig.B.13).Itisimportanttonotethatdy/dxdoesnot meandy dividedbydx,butissimplyanota-tion of the limiting process of the derivative as dened by Equation B.28.A useful expression to remember whenwhere a is a constant and nis any positive or negative number (integer or fraction), is(B.29)Ify(x)isapolynomialoralgebraicfunctionofx,weapplyEquationB.29toeach term in the polynomial and take d[constant]/dx 0. In Examples 4 through7, we evaluate the derivatives of several functions.dydx naxn1y(x) axn,y y2 y1x x2 x1dydx limx:0 yx limx:0 y(x x) y(x)xEXAMPLE 4soSubstituting this into Equation B.28 gives3ax2 bdydxdydx limx:0 yx limx:0 [3ax2 3xx x2] b bx y y(x x) y(x) a(3x2x 3xx2 x3)Suppose y(x) (that is, y as a function of x) is given bywhere a and b are constants. Then it follows that b(x x) c y(x x) a(x3 3x2x 3xx2 x3) b(x x) c y(x x) a(x x)3 y(x) ax3 bx cEXAMPLE 540x4 12x2 2dydxSolution ApplyingEquationB.29toeachtermindepen-dently,andrememberingthatd/dx (constant) 0,wehavedydx 8(5)x4 4(3)x2 2(1)x0 0y(x) 8x5 4x3 2x 7Special Properties of the DerivativeA. Derivative of the product of two functions If a function f (x) is given by theproduct of two functions, say, g(x) and h(x), then the derivative of f(x) is denedas(B.30)ddx f (x) ddx[g(x)h(x)] g dhdx h dgdxyy2y1x1x2xxyFigure B.13B.7 Intergral Calculus A.27B. Derivative of the sum of two functions If a function f (x) is equal to the sumof two functions, then the derivative of the sum is equal to the sum of the deriv-atives:(B.31)C. Chainruleofdifferentialcalculus Ify f (x) andx g(z),thendy/dz canbe written as the product of two derivatives:(B.32)D. The second derivative The second derivative of y with respect to x is denedasthederivativeofthefunctiondy/dx (thederivativeofthederivative).Itisusually written(B.33)d2ydx2ddx dydx dydzdydx dxdzddx f (x) ddx[g(x) h(x)] dgdxdhdxEXAMPLE 6dydx3x2(x 1)22x3(x 1)3 (x 1)23x2 x3(2)(x 1)3Find the derivative of y(x) x3/(x 1)2with respect to x.Solution We can rewrite this function as y(x) x3(x 1)2and apply Equation B.30:dydx (x 1)2 ddx (x3) x3 ddx (x 1)2EXAMPLE 7 h dgdx g dhdxh2 gh2 dhdx h1 dgdx ddx gh ddx (gh1) g ddx (h1) h1 ddx (g)A useful formula that follows from Equation B.30 is the deriv-ative of the quotient of two functions. Show thatSolution We can write the quotient as gh1and then applyEquations B.29 and B.30:ddx g(x)h(x) h dgdx g dhdxh2SomeofthemorecommonlyusedderivativesoffunctionsarelistedinTableB.4.INTEGRAL CALCULUSWe think of integration as the inverse of differentiation. As an example, considerthe expression(B.34)which was the result of differentiating the functiony(x) ax3 bx cf(x) dydx 3ax2 bB.7A.28 APPE NDI X Bin Example 4. We can write Equation B.34 asand ob-tainy(x)bysummingoverallvaluesofx.Mathematically,wewritethisinverseoperationFor the function f (x) given by Equation B.34, we havewhere c is a constant of the integration. This type of integral is called an indeniteintegral because its value depends on the choice of c.A general indenite integral I(x) is dened as(B.35)where f (x) is called the integrand and For a general continuous function f (x), the integral can be described as the areaunder the curve bounded by f (x) and the x axis, between two specied values of x,say, x1and x2, as in Figure B.14.TheareaoftheblueelementisapproximatelyIfwesumallthesearea elements from x1and x2and take the limit of this sum aswe obtainthe true area under the curve bounded by f (x) and x, between the limits x1and x2:(B.36)Integrals of the type dened by Equation B.36 are called denite integrals.Area limxi:0 i f (xi) xi x2x1 f(x) dxxi:0,f (xi)xi .f(x) dI(x)dx .I(x) f(x) dxy(x) (3ax2 b) dx ax3 bx cy(x) f(x) dxdy f (x) dx (3ax2 b) dxxix2f(xi)f(x)x1Figure B.14One common integral that arises in practical situations has the form(B.37)This result is obvious, being that differentiation of the right-hand side with respectto x givesdirectly. If the limits of the integration are known, this integralbecomes a denite integral and is written(B.38) x2x1 xn dx x2n1 x1n1n 1(n1)f(x) xn xn dx xn1n 1 c(n1)TABLEB.4Derivatives for SeveralFunctionsNote: The letters a and n are con-stants.ddx (ln ax) 1xddx (csc x) cot x csc xddx (sec x) tan x sec xddx (cot ax) a csc2 axddx (tan ax) a sec2 axddx (cos ax) a sin axddx (sin ax) a cos axddx (eax) aeaxddx (axn) naxn1ddx (a) 0B.7 Intergral Calculus A.29EXAMPLES1.2.3.Partial IntegrationSometimesitisusefultoapplythemethodofpartialintegration (alsocalledinte-grating by parts) to evaluate certain integrals. The method uses the property that(B.39)where u and v are carefully chosen so as to reduce a complex integral to a simplerone. In many cases, several reductions have to be made. Consider the functionThiscanbeevaluatedbyintegratingbypartstwice.First,ifwechoose we getNow, in the second term, choosewhich givesorThe Perfect DifferentialAnotherusefulmethodtorememberistheuseoftheperfectdifferential, inwhichwe look for a change of variable such that the differential of the function is the dif-ferentialoftheindependentvariableappearingintheintegrand.Forexample,consider the integralThis becomes easy to evaluate if we rewrite the differential as The integral then becomesIf we now change variables, lettingwe obtain cos2 x sin x dx y2dy y33 c cos3 x3 cy cos x, cos2 x sin x dx cos2 x d(cos x)d(cos x) sin x dx.I(x) cos2 x sin x dx x2ex dx x2ex 2xex 2ex c2 x2ex dx x2ex 2xex 2 ex dx c1v ex, u x, x2ex dx x2 d(ex) x2ex 2 exx dx c1v ex,u x2,I(x) x2ex dx u dv uv v du53 x dx x22 5352 322 8b0 x3/2 dx x5/25/2 b025 b5/2a0 x2 dx x33 a0a33A.30 APPE NDI X BTableB.5listssomeusefulindeniteintegrals.TableB.6givesGausssproba-bilityintegralandotherdeniteintegrals.Amorecompletelistcanbefoundinvarious handbooks, such as The Handbook of Chemistry and Physics, CRC Press.TABLEB.5 Some Indenite Integrals (An arbitrary constant should be added to each of these integrals.) x dx(x2 a2)3/2 1!x2 a2 eax dx 1a eax dx(x2 a2)3/2xa2!x2 a2 x(!x2 a2) dx 13 (x2 a2)3/2 cos1 ax dx x(cos1 ax) !1 a2x2a !x2 a2 dx 12 [x!x2 a2 a2 ln(x !x2 a2)] sin1 ax dx x(sin1 ax) !1 a2x2a x!a2 x2 dx 13 (a2 x2)3/2 cot2 ax dx 1a (cot ax) x !a2 x2 dx 12 x!a2 x2 a2 sin1 xa tan2 ax dx 1a (tan ax) x x dx!x2 a2 !x2 a2 dxcos2 ax1a tan ax x dx!a2 x2 !a2 x2 dxsin2 ax 1a cot ax dx!x2 a2 ln(x !x2 a2) cos2 ax dx x2sin 2 ax4a dx!a2 x2 sin1 xa cos1 xa (a2 x2 0) sin2 ax dx x2sin 2 ax4a x dxa2 x2 12 ln(a2 x2) csc ax dx 1a ln(csc ax cot ax) 1a lntan ax2 dxx2 a212a ln x ax a(x2 a2 0) sec ax dx 1a ln(sec ax tan ax) 1a lntan ax24 dxa2 x212a ln a xa x(a2 x2 0) cot ax dx 1a ln(sin ax) dxa2 x21a tan1 xa tan ax dx 1a ln(cos ax) 1a ln(sec ax) dx(a bx)2 1b(a bx) cos ax dx 1a sin ax dxx(x a) 1a ln x ax sin ax dx 1a cos ax x dxa bxxbab2ln(a bx) dxa becxxa1ac ln(a becx) dxa bx1b ln(a bx) xeax dx eaxa2 (ax 1) dxx x1 dx ln x ln ax dx (x ln ax) x xn dx xn1n 1(provided n1)B.7 Intergral Calculus A.31TABLEB.6 Gausss Probability Integral and OtherDenite Integrals(Gausss probability integral)I2n1 (1)n dndan I1I2n (1)n dndan I0I5 0 x5eax2 dx d2I1da21a3I4 0 x4eax2 dx d2I0da238 !a5I3 0 x3eax2 dx dI1da12a2I2 0 x2eax2 dx dI0da14 !a3I1 0 xeax2 dx 12aI0 0 eax2 dx 12 !a 0 xn eax dx n!an1