2518:
2504:
1968:
1639:
1069:, as he showed that the indefinite integral of a rational function is a rational function and a finite number of constant multiples of logarithms of rational functions . The algorithm suggested by Laplace is usually described in calculus textbooks; as a computer program, it was finally implemented in the 1960s.
1513:
2171:
2491:
matrix algorithm (or any algorithm that can compute the nullspace of a matrix), which is also necessary for many parts of the Risch algorithm. Gaussian elimination will produce incorrect results if it cannot correctly determine if a pivot is identically zero.
1030:
as an indefinite integral, and if it does, for determining that indefinite integral. However, the algorithm does not always succeed in identifying whether or not the antiderivative of a given function in fact can be expressed in terms of elementary functions.
2439:; this algorithm will fail if it cannot correctly determine whether coefficients vanish identically. Virtually every non-trivial algorithm relating to polynomials uses the polynomial division algorithm, the Risch algorithm included. If the constant field is
1342:
1963:{\displaystyle {\begin{aligned}F(x)=-{\frac {1}{8}}\ln &\,{\Big (}(x^{6}+15x^{4}-80x^{3}+27x^{2}-528x+781){\sqrt {x^{4}+10x^{2}-96x-71}}{\Big .}\\&{}-{\Big .}(x^{8}+20x^{6}-128x^{5}+54x^{4}-1408x^{3}+3124x^{2}+10001){\Big )}+C.\end{aligned}}}
2400:, by Manuel Bronstein, and is now being developed in Axiom's fork, FriCAS. However, the implementation did not include some of the branches for special cases completely. Currently, there is no known full implementation of the Risch algorithm.
2346:
1359:
1628:
1199:
120:
1644:
2351:
Some
Davenport "theorems" are still being clarified. For example in 2020 a counterexample to such a "theorem" was found, where it turns out that an elementary antiderivative exists after all.
2010:
2222:
1238:
1989:), which are outside the scope of the Risch algorithm. For example, Mathematica returns a result with the functions EllipticPi and EllipticF. This integral was solved by
2234:
2359:
Transforming Risch's theoretical algorithm into an algorithm that can be effectively executed by a computer was a complex task which took a long time.
3094:
1534:
Finding an elementary antiderivative is very sensitive to details. For instance, the following algebraic function (posted to sci.math.symbolic by
2362:
The case of the purely transcendental functions (which do not involve roots of polynomials) is relatively easy and was implemented early in most
1207:
The intuition for the Risch algorithm comes from the behavior of the exponential and logarithm functions under differentiation. For the function
2874:
1508:{\displaystyle \left(f\cdot (\ln g)^{n}\right)^{\prime }=f^{\prime }\left(\ln g\right)^{n}+nf{\frac {g^{\prime }}{g}}\left(\ln g\right)^{n-1}}
3133:
2551:
2449:, the problem of zero-equivalence is decidable, then the Risch algorithm is a complete algorithm. Examples of computable constant fields are
1057:. These are functions obtained by composing exponentials, logarithms, radicals, trigonometric functions, and the four arithmetic operations (
1075:
formulated the problem that is solved by the Risch algorithm. Liouville proved by analytical means that if there is an elementary solution
1045:
Some significant progress has been made in computing the logarithmic part of a mixed transcendental-algebraic integral by Brian L. Miller.
216:
1973:
But if the constant term 71 is changed to 72, it is not possible to represent the antiderivative in terms of elementary functions, as
1548:
3499:
3404:
3009:
1127:
2754:
3409:
3384:
482:
457:
3482:
3234:
2166:{\displaystyle f(x)={\frac {x^{2}+2x+1+(3x+1){\sqrt {x+\ln x}}}{x\,{\sqrt {x+\ln x}}\left(x+{\sqrt {x+\ln x}}\right)}}.}
3314:
1039:
958:
521:
2730:(in French). University of California Berkeley. St. Petersbourg, Commissionaires de l'Academie imperiale des sciences.
39:
3477:
3399:
3051:
3001:
2981:
477:
195:
2663:
3414:
3379:
2378:
462:
3394:
3065:
2931:
2531:
2397:
798:
472:
447:
129:
2228:
can solve it while FriCAS fails with "implementation incomplete (constant residues)" error in Risch algorithm):
3329:
3289:
3614:
3389:
3160:
2849:
2616:
2436:
1535:
1337:{\displaystyle \left(f\cdot e^{g}\right)^{\prime }=\left(f^{\prime }+f\cdot g^{\prime }\right)\cdot e^{g},\,}
580:
527:
408:
2179:
2176:
In fact, the antiderivative of this function has a fairly short form that can be found using substitution
1353:
were in the result of an indefinite integration, it should be expected to be inside the integral. Also, as
234:
206:
3568:
317:
3553:
3352:
2955:
2509:
2416:), in particular in the constant field. For expressions that involve only functions commonly taken to be
1542:
since version 13 shows (however, Mathematica does not use the Risch algorithm to compute this integral):
1204:
Risch developed a method that allows one to consider only a finite set of functions of
Liouville's form.
831:
439:
277:
249:
2630:
3609:
3522:
3489:
3357:
2900:
2385:, though for simplicity it could only deal with square roots and repeated square roots and not general
702:
666:
443:
322:
211:
201:
2412:
because it needs to check, as a part of its operation, if certain expressions are equivalent to zero (
3604:
2541:
1994:
466:
2594:
302:
3367:
601:
161:
1526:
were in the result of an integration, then only a few powers of the logarithm should be expected.
3227:
2429:
2421:
2363:
2001:
1978:
1229:
980:
915:
707:
596:
2556:
2536:
1834:
1816:
999:
951:
880:
841:
725:
661:
585:
3527:
3429:
925:
591:
362:
307:
268:
174:
2341:{\displaystyle F(x)=2\left({\sqrt {x+\ln x}}+\ln \left(x+{\sqrt {x+\ln x}}\right)\right)+C.}
3374:
3264:
3031:
2561:
2488:
1062:
1019:
1003:
972:
930:
910:
836:
505:
424:
398:
312:
1006:. It is based on the form of the function being integrated and on methods for integrating
8:
3509:
3424:
3419:
3309:
2417:
1993:(and in what cases it is elementary), but the strict proof for it was ultimately done by
1539:
1054:
1027:
905:
875:
865:
752:
606:
403:
259:
142:
137:
3035:
3532:
3469:
3362:
3334:
3299:
3220:
3177:
3113:
3025:
2993:
2794:
2706:
2546:
2523:
2396:
The general case was solved and almost fully implemented in
Scratchpad, a precursor of
2390:
2386:
2382:
1066:
1023:
988:
870:
773:
757:
697:
692:
687:
651:
532:
451:
357:
352:
156:
151:
2944:
2408:
The Risch algorithm applied to general elementary functions is not an algorithm but a
3581:
3558:
3494:
3464:
3456:
3434:
3304:
3047:
3005:
2977:
2798:
2786:
2710:
2698:
2517:
1990:
1986:
1007:
944:
778:
556:
434:
387:
244:
239:
3198:
3147:
3128:
2428:
to the list of elementary functions, it is known that no such algorithm exists; see
2420:
it is not known whether an algorithm performing such a check exists or not (current
3576:
3439:
3324:
3284:
3279:
3274:
3269:
3259:
3169:
3142:
3103:
3074:
3039:
2940:
2776:
2766:
2690:
2413:
1072:
788:
682:
656:
517:
429:
393:
2875:"integration - Does there exist a complete implementation of the Risch algorithm?"
3563:
3319:
2771:
2409:
920:
793:
747:
742:
629:
542:
3517:
2425:
984:
803:
611:
378:
3078:
1038:
is a simpler, faster, but less powerful variant that was developed in 1976 by
3598:
3548:
2790:
2702:
783:
547:
297:
254:
2595:"On the integration of elementary functions: Computing the logarithmic part"
3294:
3021:
2725:
537:
282:
2963:
ISSAC'98, Rostock (August 1998) and
Differential Algebra Workshop, Rutgers
1034:
The complete description of the Risch algorithm takes over 100 pages. The
2000:
The following is a more complex example that involves both algebraic and
900:
2781:
3181:
3117:
2694:
2440:
2367:
646:
570:
292:
287:
191:
2755:"Torsion points, Pell's equation, and integration in elementary terms"
3203:
2678:
2377:
The case of purely algebraic functions was solved and implemented in
1015:
995:
575:
565:
3173:
3108:
3089:
3043:
2503:
3243:
1011:
641:
383:
340:
29:
2929:
Bronstein, Manuel (1990). "Integration of elementary functions".
2371:
1026:, because it is a method for deciding whether a function has an
2655:
1974:
1623:{\displaystyle f(x)={\frac {x}{\sqrt {x^{4}+10x^{2}-96x-71}}},}
3030:. Boston, MA: Kluwer Academic Publishers. pp. xxii+585.
2225:
991:, a specialist in computer algebra who developed it in 1968.
3212:
3129:"The solution of the problem of integration in finite terms"
3158:
Rosenlicht, Maxwell (1972). "Integration in finite terms".
3195:
2472:
with rational number coefficients, respectively, where
1194:{\displaystyle g=v+\sum _{i<n}\alpha _{i}\ln(u_{i})}
2850:"Manuel Bronstein on Axiom's Integration Capabilities"
2237:
2182:
2013:
1642:
1551:
1362:
1241:
1130:
42:
2499:
3168:(9). Mathematical Association of America: 963â972.
3063:Moses, Joel (2012). "Macsyma: A personal history".
2654:This example was posted by Manuel Bronstein to the
2466:, i.e., rational numbers and rational functions in
979:is a method of indefinite integration used in some
3020:
2580:
2340:
2216:
2165:
1962:
1622:
1507:
1336:
1193:
114:
3095:Transactions of the American Mathematical Society
2901:"Mathematica 7 Documentation: PolynomialQuotient"
2753:Masser, David; Zannier, Umberto (December 2020).
1942:
1685:
3596:
115:{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}
2679:"Sur la mĂŠthode d'intĂŠgration de M. TchĂŠbychef"
987:. It is named after the American mathematician
2723:
1981:may here return an antiderivative in terms of
1538:in 1993) has an elementary antiderivative, as
3228:
3134:Bulletin of the American Mathematical Society
3024:; Czapor, Stephen R.; Labahn, George (1992).
2752:
2374:soon after the publication of Risch's paper.
952:
3090:"The problem of integration in finite terms"
2478:is an indeterminate that does not depend on
2617:"A Christmas present for your favorite CAS"
2424:use heuristics); moreover, if one adds the
3235:
3221:
3157:
3102:. American Mathematical Society: 167â189.
2676:
2552:Liouville's theorem (differential algebra)
959:
945:
18:Method for evaluating indefinite integrals
3146:
3107:
2998:On the integration of algebraic functions
2992:
2971:
2953:
2928:
2847:
2835:
2823:
2780:
2770:
2740:
2102:
1682:
1333:
1053:The Risch algorithm is used to integrate
75:
3500:Common integrals in quantum field theory
2435:Note that this issue also arises in the
3410:Differentiation under the integral sign
2848:Bronstein, Manuel (September 5, 2003).
2366:. The first implementation was done by
483:Differentiating under the integral sign
3597:
2592:
2443:, i.e., for elements not dependent on
1121:such that the solution is of the form
3216:
3196:
3126:
3087:
3062:
2811:
2614:
2217:{\displaystyle u=x+{\sqrt {x+\ln x}}}
2672:
2670:
1065:solved this problem for the case of
2677:Zolotareff, G. (December 1, 1872).
1529:
13:
2615:Cohen, Henri (December 21, 1993).
1463:
1416:
1403:
1307:
1288:
1270:
24:Part of a series of articles about
14:
3626:
3189:
3002:Lecture Notes in Computer Science
2667:
2354:
2581:Geddes, Czapor & Labahn 1992
2516:
2502:
3148:10.1090/S0002-9904-1970-12454-5
3066:Journal of Symbolic Computation
3027:Algorithms for computer algebra
2956:"Symbolic Integration Tutorial"
2932:Journal of Symbolic Computation
2893:
2867:
2841:
2829:
2817:
2805:
2532:Axiom (computer algebra system)
2403:
2746:
2734:
2724:Chebyshev, P. L. (1899â1907).
2717:
2648:
2623:
2608:
2586:
2574:
2247:
2241:
2075:
2060:
2023:
2017:
1937:
1838:
1766:
1690:
1656:
1650:
1561:
1555:
1388:
1375:
1188:
1175:
1048:
109:
103:
94:
88:
72:
66:
1:
3242:
3161:American Mathematical Monthly
2945:10.1016/s0747-7171(08)80027-2
2922:
2593:Miller, Brian L. (May 2012).
2487:This is also an issue in the
2437:polynomial division algorithm
409:Integral of inverse functions
2772:10.4310/ACTA.2020.v225.n2.a2
7:
3315:LebesgueâStieltjes integral
3004:. Vol. 102. Springer.
2510:Computer programming portal
2495:
1091:then there exist constants
832:Calculus on Euclidean space
250:Logarithmic differentiation
10:
3631:
3330:RiemannâStieltjes integral
3290:HenstockâKurzweil integral
2972:Bronstein, Manuel (2005).
2954:Bronstein, Manuel (1998).
2727:Oeuvres de P.L. Tchebychef
1115:in the field generated by
998:transforms the problem of
3569:Proof that 22/7 exceeds Ď
3541:
3508:
3455:
3343:
3250:
3079:10.1016/j.jsc.2010.08.018
2542:Incomplete gamma function
566:Summand limit (term test)
2905:Section: Possible Issues
2660:comp.soft-sys.math.maple
2567:
2422:computer algebra systems
2364:computer algebra systems
2002:transcendental functions
1979:computer algebra systems
1230:differentiable functions
981:computer algebra systems
245:Implicit differentiation
235:Differentiation notation
162:Inverse function theorem
3554:EulerâMaclaurin formula
2426:absolute value function
2389:or other non-quadratic
708:Helmholtz decomposition
3523:RussoâVallois integral
3490:BoseâEinstein integral
3405:Parametric derivatives
2974:Symbolic Integration I
2557:Nonelementary integral
2537:Closed-form expression
2342:
2218:
2167:
1964:
1624:
1509:
1338:
1195:
1036:RischâNorman algorithm
842:Limit of distributions
662:Directional derivative
318:FaĂ di Bruno's formula
116:
3528:Stratonovich integral
3474:FermiâDirac integral
3430:Numerical integration
3127:Risch, R. H. (1970).
3088:Risch, R. H. (1969).
2683:Mathematische Annalen
2662:on November 24, 2000.
2343:
2219:
2168:
1965:
1625:
1510:
1339:
1196:
1020:exponential functions
926:Mathematical analysis
837:Generalized functions
522:arithmetico-geometric
363:Leibniz integral rule
117:
3615:Differential algebra
3510:Stochastic integrals
2562:Symbolic integration
2489:Gaussian elimination
2430:Richardson's theorem
2235:
2180:
2011:
1640:
1549:
1360:
1239:
1128:
1055:elementary functions
1022:. Risch called it a
973:symbolic computation
931:Nonstandard analysis
399:Lebesgue integration
269:Rules and identities
40:
3420:Contour integration
3310:Kolmogorov integral
3036:1992afca.book.....G
2994:Davenport, James H.
2393:between variables.
2391:algebraic relations
1540:Wolfram Mathematica
1028:elementary function
602:Cauchy condensation
404:Contour integration
130:Fundamental theorem
57:
3533:Skorokhod integral
3470:Dirichlet integral
3457:Improper integrals
3400:Reduction formulas
3335:Regulated integral
3300:Hellinger integral
3197:Bhatt, Bhuvanesh.
2881:. October 15, 2020
2695:10.1007/BF01442910
2547:Lists of integrals
2524:Mathematics portal
2383:James H. Davenport
2338:
2214:
2163:
1987:elliptic integrals
1960:
1958:
1620:
1505:
1334:
1191:
1158:
1067:rational functions
1024:decision procedure
1008:rational functions
1002:into a problem in
989:Robert Henry Risch
774:Partial derivative
703:generalized Stokes
597:Alternating series
478:Reduction formulae
467:Heaviside's method
448:tangent half-angle
435:Cylindrical shells
358:Integral transform
353:Lists of integrals
157:Mean value theorem
112:
43:
3610:Integral calculus
3592:
3591:
3495:Frullani integral
3465:Gaussian integral
3415:Laplace transform
3390:Inverse functions
3380:Partial fractions
3305:Khinchin integral
3265:Lebesgue integral
3199:"Risch Algorithm"
3011:978-3-540-10290-8
2854:groups.google.com
2317:
2278:
2212:
2158:
2150:
2120:
2095:
1977:also shows. Some
1812:
1673:
1615:
1614:
1471:
1143:
969:
968:
849:
848:
811:
810:
779:Multiple integral
715:
714:
619:
618:
586:Direct comparison
557:Convergence tests
495:
494:
463:Partial fractions
330:
329:
240:Second derivative
3622:
3605:Computer algebra
3440:Trapezoidal rule
3425:Laplace's method
3325:Pfeffer integral
3285:Darboux integral
3280:Daniell integral
3275:Bochner integral
3270:Burkill integral
3260:Riemann integral
3237:
3230:
3223:
3214:
3213:
3209:
3208:
3185:
3152:
3150:
3121:
3111:
3082:
3057:
3022:Geddes, Keith O.
3015:
2987:
2966:
2960:
2948:
2916:
2915:
2913:
2911:
2897:
2891:
2890:
2888:
2886:
2871:
2865:
2864:
2862:
2860:
2845:
2839:
2833:
2827:
2821:
2815:
2809:
2803:
2802:
2784:
2774:
2759:Acta Mathematica
2750:
2744:
2738:
2732:
2731:
2721:
2715:
2714:
2674:
2665:
2652:
2646:
2645:
2643:
2641:
2627:
2621:
2620:
2612:
2606:
2605:
2603:
2601:
2590:
2584:
2578:
2526:
2521:
2520:
2512:
2507:
2506:
2483:
2477:
2471:
2465:
2454:
2448:
2414:constant problem
2347:
2345:
2344:
2339:
2328:
2324:
2323:
2319:
2318:
2301:
2279:
2262:
2223:
2221:
2220:
2215:
2213:
2196:
2172:
2170:
2169:
2164:
2159:
2157:
2156:
2152:
2151:
2134:
2121:
2104:
2097:
2096:
2079:
2041:
2040:
2030:
1985:functions (i.e.
1969:
1967:
1966:
1961:
1959:
1946:
1945:
1930:
1929:
1914:
1913:
1898:
1897:
1882:
1881:
1866:
1865:
1850:
1849:
1837:
1836:
1828:
1823:
1819:
1818:
1813:
1796:
1795:
1780:
1779:
1770:
1750:
1749:
1734:
1733:
1718:
1717:
1702:
1701:
1689:
1688:
1674:
1666:
1629:
1627:
1626:
1621:
1616:
1598:
1597:
1582:
1581:
1572:
1568:
1530:Problem examples
1525:
1514:
1512:
1511:
1506:
1504:
1503:
1492:
1488:
1472:
1467:
1466:
1457:
1446:
1445:
1440:
1436:
1420:
1419:
1407:
1406:
1401:
1397:
1396:
1395:
1352:
1343:
1341:
1340:
1335:
1329:
1328:
1316:
1312:
1311:
1310:
1292:
1291:
1274:
1273:
1268:
1264:
1263:
1262:
1227:
1221:
1215:
1200:
1198:
1197:
1192:
1187:
1186:
1168:
1167:
1157:
1120:
1114:
1108:
1099:
1090:
1081:to the equation
1080:
1060:
961:
954:
947:
895:
860:
826:
825:
822:
789:Surface integral
732:
731:
728:
636:
635:
632:
592:Limit comparison
512:
511:
508:
394:Riemann integral
347:
346:
343:
303:L'HĂ´pital's rule
260:Taylor's theorem
181:
180:
177:
121:
119:
118:
113:
65:
56:
51:
21:
20:
3630:
3629:
3625:
3624:
3623:
3621:
3620:
3619:
3595:
3594:
3593:
3588:
3564:Integration Bee
3537:
3504:
3451:
3447:Risch algorithm
3385:Euler's formula
3345:
3339:
3320:Pettis integral
3252:
3246:
3241:
3192:
3174:10.2307/2318066
3109:10.2307/1995313
3054:
3044:10.1007/b102438
3012:
2984:
2958:
2925:
2920:
2919:
2909:
2907:
2899:
2898:
2894:
2884:
2882:
2873:
2872:
2868:
2858:
2856:
2846:
2842:
2834:
2830:
2822:
2818:
2810:
2806:
2751:
2747:
2739:
2735:
2722:
2718:
2675:
2668:
2653:
2649:
2639:
2637:
2631:"Wolfram Cloud"
2629:
2628:
2624:
2613:
2609:
2599:
2597:
2591:
2587:
2579:
2575:
2570:
2522:
2515:
2508:
2501:
2498:
2479:
2473:
2467:
2456:
2450:
2444:
2406:
2357:
2300:
2293:
2289:
2261:
2260:
2256:
2236:
2233:
2232:
2195:
2181:
2178:
2177:
2133:
2126:
2122:
2103:
2098:
2078:
2036:
2032:
2031:
2029:
2012:
2009:
2008:
1957:
1956:
1941:
1940:
1925:
1921:
1909:
1905:
1893:
1889:
1877:
1873:
1861:
1857:
1845:
1841:
1833:
1832:
1827:
1821:
1820:
1815:
1814:
1791:
1787:
1775:
1771:
1769:
1745:
1741:
1729:
1725:
1713:
1709:
1697:
1693:
1684:
1683:
1678:
1665:
1643:
1641:
1638:
1637:
1593:
1589:
1577:
1573:
1567:
1550:
1547:
1546:
1532:
1519:
1493:
1478:
1474:
1473:
1462:
1458:
1456:
1441:
1426:
1422:
1421:
1415:
1411:
1402:
1391:
1387:
1368:
1364:
1363:
1361:
1358:
1357:
1348:
1324:
1320:
1306:
1302:
1287:
1283:
1282:
1278:
1269:
1258:
1254:
1247:
1243:
1242:
1240:
1237:
1236:
1223:
1217:
1208:
1182:
1178:
1163:
1159:
1147:
1129:
1126:
1125:
1116:
1110:
1106:
1101:
1097:
1092:
1082:
1076:
1058:
1051:
985:antiderivatives
977:Risch algorithm
965:
936:
935:
921:Integration Bee
896:
893:
886:
885:
861:
858:
851:
850:
823:
820:
813:
812:
794:Volume integral
729:
724:
717:
716:
633:
628:
621:
620:
590:
509:
504:
497:
496:
488:Risch algorithm
458:Euler's formula
344:
339:
332:
331:
313:General Leibniz
196:generalizations
178:
173:
166:
152:Rolle's theorem
147:
122:
58:
52:
47:
41:
38:
37:
19:
12:
11:
5:
3628:
3618:
3617:
3612:
3607:
3590:
3589:
3587:
3586:
3585:
3584:
3579:
3571:
3566:
3561:
3559:Gabriel's horn
3556:
3551:
3545:
3543:
3539:
3538:
3536:
3535:
3530:
3525:
3520:
3514:
3512:
3506:
3505:
3503:
3502:
3497:
3492:
3487:
3486:
3485:
3480:
3472:
3467:
3461:
3459:
3453:
3452:
3450:
3449:
3444:
3443:
3442:
3437:
3435:Simpson's rule
3427:
3422:
3417:
3412:
3407:
3402:
3397:
3395:Changing order
3392:
3387:
3382:
3377:
3372:
3371:
3370:
3365:
3360:
3349:
3347:
3341:
3340:
3338:
3337:
3332:
3327:
3322:
3317:
3312:
3307:
3302:
3297:
3292:
3287:
3282:
3277:
3272:
3267:
3262:
3256:
3254:
3248:
3247:
3240:
3239:
3232:
3225:
3217:
3211:
3210:
3191:
3190:External links
3188:
3187:
3186:
3154:
3153:
3141:(3): 605â608.
3123:
3122:
3084:
3083:
3073:(2): 123â130.
3059:
3058:
3052:
3017:
3016:
3010:
2989:
2988:
2982:
2968:
2967:
2950:
2949:
2939:(2): 117â173.
2924:
2921:
2918:
2917:
2892:
2866:
2840:
2836:Bronstein 1990
2828:
2824:Davenport 1981
2816:
2804:
2765:(2): 227â312.
2745:
2741:Bronstein 1998
2733:
2716:
2689:(4): 560â580.
2666:
2647:
2622:
2607:
2585:
2572:
2571:
2569:
2566:
2565:
2564:
2559:
2554:
2549:
2544:
2539:
2534:
2528:
2527:
2513:
2497:
2494:
2410:semi-algorithm
2405:
2402:
2356:
2355:Implementation
2353:
2349:
2348:
2337:
2334:
2331:
2327:
2322:
2316:
2313:
2310:
2307:
2304:
2299:
2296:
2292:
2288:
2285:
2282:
2277:
2274:
2271:
2268:
2265:
2259:
2255:
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2243:
2240:
2211:
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2205:
2202:
2199:
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2191:
2188:
2185:
2174:
2173:
2162:
2155:
2149:
2146:
2143:
2140:
2137:
2132:
2129:
2125:
2119:
2116:
2113:
2110:
2107:
2101:
2094:
2091:
2088:
2085:
2082:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
2053:
2050:
2047:
2044:
2039:
2035:
2028:
2025:
2022:
2019:
2016:
1983:non-elementary
1971:
1970:
1955:
1952:
1949:
1944:
1939:
1936:
1933:
1928:
1924:
1920:
1917:
1912:
1908:
1904:
1901:
1896:
1892:
1888:
1885:
1880:
1876:
1872:
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1721:
1716:
1712:
1708:
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1619:
1613:
1610:
1607:
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1601:
1596:
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1499:
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1487:
1484:
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1477:
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1455:
1452:
1449:
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1439:
1435:
1432:
1429:
1425:
1418:
1414:
1410:
1405:
1400:
1394:
1390:
1386:
1383:
1380:
1377:
1374:
1371:
1367:
1345:
1344:
1332:
1327:
1323:
1319:
1315:
1309:
1305:
1301:
1298:
1295:
1290:
1286:
1281:
1277:
1272:
1267:
1261:
1257:
1253:
1250:
1246:
1202:
1201:
1190:
1185:
1181:
1177:
1174:
1171:
1166:
1162:
1156:
1153:
1150:
1146:
1142:
1139:
1136:
1133:
1104:
1100:and functions
1095:
1050:
1047:
967:
966:
964:
963:
956:
949:
941:
938:
937:
934:
933:
928:
923:
918:
916:List of topics
913:
908:
903:
897:
892:
891:
888:
887:
884:
883:
878:
873:
868:
862:
857:
856:
853:
852:
847:
846:
845:
844:
839:
834:
824:
819:
818:
815:
814:
809:
808:
807:
806:
801:
796:
791:
786:
781:
776:
768:
767:
763:
762:
761:
760:
755:
750:
745:
737:
736:
730:
723:
722:
719:
718:
713:
712:
711:
710:
705:
700:
695:
690:
685:
677:
676:
672:
671:
670:
669:
664:
659:
654:
649:
644:
634:
627:
626:
623:
622:
617:
616:
615:
614:
609:
604:
599:
594:
588:
583:
578:
573:
568:
560:
559:
553:
552:
551:
550:
545:
540:
535:
530:
525:
510:
503:
502:
499:
498:
493:
492:
491:
490:
485:
480:
475:
473:Changing order
470:
460:
455:
437:
432:
427:
419:
418:
417:Integration by
414:
413:
412:
411:
406:
401:
396:
391:
381:
379:Antiderivative
373:
372:
368:
367:
366:
365:
360:
355:
345:
338:
337:
334:
333:
328:
327:
326:
325:
320:
315:
310:
305:
300:
295:
290:
285:
280:
272:
271:
265:
264:
263:
262:
257:
252:
247:
242:
237:
229:
228:
224:
223:
222:
221:
220:
219:
214:
209:
199:
186:
185:
179:
172:
171:
168:
167:
165:
164:
159:
154:
148:
146:
145:
140:
134:
133:
132:
124:
123:
111:
108:
105:
102:
99:
96:
93:
90:
87:
84:
81:
78:
74:
71:
68:
64:
61:
55:
50:
46:
36:
33:
32:
26:
25:
17:
9:
6:
4:
3:
2:
3627:
3616:
3613:
3611:
3608:
3606:
3603:
3602:
3600:
3583:
3580:
3578:
3575:
3574:
3572:
3570:
3567:
3565:
3562:
3560:
3557:
3555:
3552:
3550:
3549:Basel problem
3547:
3546:
3544:
3542:Miscellaneous
3540:
3534:
3531:
3529:
3526:
3524:
3521:
3519:
3516:
3515:
3513:
3511:
3507:
3501:
3498:
3496:
3493:
3491:
3488:
3484:
3481:
3479:
3476:
3475:
3473:
3471:
3468:
3466:
3463:
3462:
3460:
3458:
3454:
3448:
3445:
3441:
3438:
3436:
3433:
3432:
3431:
3428:
3426:
3423:
3421:
3418:
3416:
3413:
3411:
3408:
3406:
3403:
3401:
3398:
3396:
3393:
3391:
3388:
3386:
3383:
3381:
3378:
3376:
3373:
3369:
3366:
3364:
3361:
3359:
3358:Trigonometric
3356:
3355:
3354:
3351:
3350:
3348:
3342:
3336:
3333:
3331:
3328:
3326:
3323:
3321:
3318:
3316:
3313:
3311:
3308:
3306:
3303:
3301:
3298:
3296:
3295:Haar integral
3293:
3291:
3288:
3286:
3283:
3281:
3278:
3276:
3273:
3271:
3268:
3266:
3263:
3261:
3258:
3257:
3255:
3249:
3245:
3238:
3233:
3231:
3226:
3224:
3219:
3218:
3215:
3206:
3205:
3200:
3194:
3193:
3183:
3179:
3175:
3171:
3167:
3163:
3162:
3156:
3155:
3149:
3144:
3140:
3136:
3135:
3130:
3125:
3124:
3119:
3115:
3110:
3105:
3101:
3097:
3096:
3091:
3086:
3085:
3080:
3076:
3072:
3068:
3067:
3061:
3060:
3055:
3053:0-7923-9259-0
3049:
3045:
3041:
3037:
3033:
3029:
3028:
3023:
3019:
3018:
3013:
3007:
3003:
2999:
2995:
2991:
2990:
2985:
2983:3-540-21493-3
2979:
2975:
2970:
2969:
2964:
2957:
2952:
2951:
2946:
2942:
2938:
2934:
2933:
2927:
2926:
2906:
2902:
2896:
2880:
2876:
2870:
2855:
2851:
2844:
2837:
2832:
2825:
2820:
2813:
2808:
2800:
2796:
2792:
2788:
2783:
2778:
2773:
2768:
2764:
2760:
2756:
2749:
2742:
2737:
2729:
2728:
2720:
2712:
2708:
2704:
2700:
2696:
2692:
2688:
2685:(in French).
2684:
2680:
2673:
2671:
2664:
2661:
2657:
2651:
2636:
2635:Wolfram Cloud
2632:
2626:
2618:
2611:
2596:
2589:
2582:
2577:
2573:
2563:
2560:
2558:
2555:
2553:
2550:
2548:
2545:
2543:
2540:
2538:
2535:
2533:
2530:
2529:
2525:
2519:
2514:
2511:
2505:
2500:
2493:
2490:
2485:
2482:
2476:
2470:
2463:
2459:
2453:
2447:
2442:
2438:
2433:
2431:
2427:
2423:
2419:
2415:
2411:
2401:
2399:
2394:
2392:
2388:
2384:
2380:
2375:
2373:
2369:
2365:
2360:
2352:
2335:
2332:
2329:
2325:
2320:
2314:
2311:
2308:
2305:
2302:
2297:
2294:
2290:
2286:
2283:
2280:
2275:
2272:
2269:
2266:
2263:
2257:
2253:
2250:
2244:
2238:
2231:
2230:
2229:
2227:
2209:
2206:
2203:
2200:
2197:
2192:
2189:
2186:
2183:
2160:
2153:
2147:
2144:
2141:
2138:
2135:
2130:
2127:
2123:
2117:
2114:
2111:
2108:
2105:
2099:
2092:
2089:
2086:
2083:
2080:
2072:
2069:
2066:
2063:
2057:
2054:
2051:
2048:
2045:
2042:
2037:
2033:
2026:
2020:
2014:
2007:
2006:
2005:
2003:
1998:
1996:
1992:
1988:
1984:
1980:
1976:
1953:
1950:
1947:
1934:
1931:
1926:
1922:
1918:
1915:
1910:
1906:
1902:
1899:
1894:
1890:
1886:
1883:
1878:
1874:
1870:
1867:
1862:
1858:
1854:
1851:
1846:
1842:
1829:
1825:
1809:
1806:
1803:
1800:
1797:
1792:
1788:
1784:
1781:
1776:
1772:
1763:
1760:
1757:
1754:
1751:
1746:
1742:
1738:
1735:
1730:
1726:
1722:
1719:
1714:
1710:
1706:
1703:
1698:
1694:
1680:
1675:
1670:
1667:
1662:
1659:
1653:
1647:
1636:
1635:
1634:
1617:
1611:
1608:
1605:
1602:
1599:
1594:
1590:
1586:
1583:
1578:
1574:
1569:
1564:
1558:
1552:
1545:
1544:
1543:
1541:
1537:
1527:
1523:
1500:
1497:
1494:
1489:
1485:
1482:
1479:
1475:
1468:
1459:
1453:
1450:
1447:
1442:
1437:
1433:
1430:
1427:
1423:
1412:
1408:
1398:
1392:
1384:
1381:
1378:
1372:
1369:
1365:
1356:
1355:
1354:
1351:
1330:
1325:
1321:
1317:
1313:
1303:
1299:
1296:
1293:
1284:
1279:
1275:
1265:
1259:
1255:
1251:
1248:
1244:
1235:
1234:
1233:
1231:
1226:
1220:
1214:
1211:
1205:
1183:
1179:
1172:
1169:
1164:
1160:
1154:
1151:
1148:
1144:
1140:
1137:
1134:
1131:
1124:
1123:
1122:
1119:
1113:
1107:
1098:
1089:
1085:
1079:
1074:
1070:
1068:
1064:
1056:
1046:
1043:
1041:
1040:Arthur Norman
1037:
1032:
1029:
1025:
1021:
1017:
1013:
1009:
1005:
1001:
997:
992:
990:
986:
982:
978:
974:
962:
957:
955:
950:
948:
943:
942:
940:
939:
932:
929:
927:
924:
922:
919:
917:
914:
912:
909:
907:
904:
902:
899:
898:
890:
889:
882:
879:
877:
874:
872:
869:
867:
864:
863:
855:
854:
843:
840:
838:
835:
833:
830:
829:
828:
827:
817:
816:
805:
802:
800:
797:
795:
792:
790:
787:
785:
784:Line integral
782:
780:
777:
775:
772:
771:
770:
769:
765:
764:
759:
756:
754:
751:
749:
746:
744:
741:
740:
739:
738:
734:
733:
727:
726:Multivariable
721:
720:
709:
706:
704:
701:
699:
696:
694:
691:
689:
686:
684:
681:
680:
679:
678:
674:
673:
668:
665:
663:
660:
658:
655:
653:
650:
648:
645:
643:
640:
639:
638:
637:
631:
625:
624:
613:
610:
608:
605:
603:
600:
598:
595:
593:
589:
587:
584:
582:
579:
577:
574:
572:
569:
567:
564:
563:
562:
561:
558:
555:
554:
549:
546:
544:
541:
539:
536:
534:
531:
529:
526:
523:
519:
516:
515:
514:
513:
507:
501:
500:
489:
486:
484:
481:
479:
476:
474:
471:
468:
464:
461:
459:
456:
453:
449:
445:
444:trigonometric
441:
438:
436:
433:
431:
428:
426:
423:
422:
421:
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3344:Integration
3202:
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2879:MathOverflow
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2859:February 10,
2857:. Retrieved
2853:
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2831:
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2807:
2782:11384/110046
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2640:December 11,
2638:. Retrieved
2634:
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2600:December 10,
2598:. Retrieved
2588:
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2404:Decidability
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1999:
1982:
1972:
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440:Substitution
202:Differential
175:Differential
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3368:Weierstrass
1536:Henri Cohen
1049:Description
1000:integration
901:Precalculus
894:Miscellanea
859:Specialized
766:Definitions
533:Alternating
371:Definitions
184:Definitions
3599:Categories
3483:incomplete
3346:techniques
2923:References
2812:Moses 2012
2441:computable
2418:elementary
2368:Joel Moses
1232:, we have
1016:logarithms
881:Variations
876:Stochastic
866:Fractional
735:Formalisms
698:Divergence
667:Identities
647:Divergence
192:Derivative
143:Continuity
3253:integrals
3251:Types of
3244:Integrals
3204:MathWorld
2799:221405883
2791:1871-2509
2711:123629827
2703:1432-1807
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1991:Chebyshev
1900:−
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1600:−
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1073:Liouville
996:algorithm
871:Malliavin
758:Geometric
657:Laplacian
607:Dirichlet
518:Geometric
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3573:Volumes
3478:complete
3375:By parts
2996:(1981).
2910:July 17,
2496:See also
2387:Radicals
1633:namely:
1518:then if
1216:, where
1012:radicals
983:to find
911:Glossary
821:Advanced
799:Jacobian
753:Exterior
683:Gradient
675:Theorems
642:Gradient
581:Integral
543:Binomial
528:Harmonic
388:improper
384:Integral
341:Integral
323:Reynolds
298:Quotient
227:Concepts
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3577:Washers
3182:2318066
3118:1995313
3032:Bibcode
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1063:Laplace
1059:+ â à á
1004:algebra
906:History
804:Hessian
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2379:Reduce
1975:FriCAS
1347:so if
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743:Matrix
630:Vector
548:Taylor
506:Series
138:Limits
3363:Euler
3178:JSTOR
3114:JSTOR
2959:(PDF)
2795:S2CID
2707:S2CID
2568:Notes
2398:Axiom
2226:SymPy
1935:10001
571:Ratio
538:Power
452:Euler
430:Discs
425:Parts
293:Power
288:Chain
217:total
3048:ISBN
3006:ISBN
2978:ISBN
2912:2010
2887:2023
2861:2023
2787:ISSN
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