25:
2734:
2137:
1962:
2254:
2524:
816:
1630:
axis about the origin, ω stays the same. That is, ω acts like an even function. This is the same as the symmetry of the cosine, which is an even function, so the mnemonic tells us to use the substitution
2578:
1442:
2324:
1307:
902:
2028:
546:
461:
1796:
1125:
2368:
is even, the integrand as a whole ω is odd, so it does not fall under rule 1. It also lacks the symmetries described in rules 2 and 3, so we fall back to the last-resort substitution
1184:
376:
2588:
295:
2786:
1578:
2362:
1740:
244:
1521:
2412:
1624:
665:
1345:
955:
1222:
1053:
1004:
616:
2038:
1833:
1690:
1661:
578:
493:
408:
2756:
1475:
205:
179:
1843:
149:
2147:
54:
318:
324:
axis. The translations and reflections are ones that correspond to the symmetries and periodicities of the basic trigonometric functions.
2420:
721:
1692:. The integrand involving transcendental functions has been reduced to one involving a rational function (a constant). The result is
76:
2529:
823:
1350:
2827:
2274:
1227:
1477:
allows one to reduce to a rational function, this last change of variable being most interesting in the fourth case (
837:
2822:
1977:
37:
501:
416:
1758:
1074:
47:
41:
33:
2729:{\displaystyle \int {\frac {\mathrm {d} t}{1+\beta \cos t}}={\frac {2}{\sqrt {1-\beta ^{2}}}}\arctan \left+c.}
1133:
334:
58:
249:
2780:
1544:
2334:
1695:
93:
210:
104:
1480:
2371:
2132:{\displaystyle {\frac {1}{1-u^{2}}}={\frac {1}{2}}\left({\frac {1}{1+u}}+{\frac {1}{1-u}}\right)}
1588:
624:
1312:
919:
1189:
1020:
971:
586:
1806:
1666:
1634:
819:
551:
466:
381:
152:
1957:{\displaystyle {\frac {dt}{\sin t}}=-{\frac {du}{\sin ^{2}t}}=-{\frac {du}{\ 1-\cos ^{2}t}}.}
1447:
2249:{\displaystyle \int {\frac {dt}{\sin t}}=-{\frac {1}{2}}\ln {\frac {1+\cos t}{1-\cos t}}+c.}
184:
158:
125:
2816:
1059:
714:: one shows that the proposed change of variable reduces (if the rule applies and if
98:
300:
1803:
has the same symmetries as the one in example 1, so we use the same substitution
1742:, which is of course elementary and could have been done without Bioche's rules.
108:
1309:), in the case of hyperbolic sine and cosine, a good change of variable is
583:
If two of the preceding relations both hold, a good change of variables is
695:
has a mnemonic advantage, which is that we choose the change of variables
16:
Aids the computation of indefinite integrals involving sines and cosines
711:
2807:
How to
Integrate It: A practical guide to finding elementary integrals
116:
2519:{\displaystyle \cos t={\frac {1-\tan ^{2}(t/2)}{1+\tan ^{2}(t/2)}}}
297:. We consider the behavior of this entire integrand, including the
811:{\displaystyle f(t)={\frac {P(\sin t,\cos t)}{Q(\sin t,\cos t)}}}
112:
18:
2767:(in French): 1–2. Archived from the original on 18 July 2022
103:(1859–1949), are rules to aid in the computation of certain
687:
by a sign. Although the rules could be stated in terms of
1663:(rule 1). Under this substitution, the integral becomes
2573:{\displaystyle v={\sqrt {\frac {1-\beta }{1+\beta }}}u}
2035:
which can be integrated using partial fractions, since
1437:{\displaystyle \sinh(t),\tanh(t),\cosh(2t),\tanh(t/2)}
303:
2591:
2532:
2423:
2374:
2337:
2277:
2150:
2041:
1980:
1846:
1809:
1761:
1698:
1669:
1637:
1591:
1547:
1483:
1450:
1353:
1315:
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1192:
1136:
1077:
1023:
974:
922:
840:
724:
627:
589:
554:
504:
469:
419:
384:
337:
252:
213:
187:
161:
128:
2728:
2572:
2518:
2406:
2356:
2319:{\displaystyle \int {\frac {dt}{1+\beta \cos t}},}
2318:
2248:
2131:
2022:
1956:
1827:
1790:
1734:
1684:
1655:
1626:is an odd function, but under a reflection of the
1618:
1572:
1515:
1469:
1436:
1339:
1301:
1216:
1178:
1119:
1047:
998:
949:
896:
810:
659:
610:
572:
540:
487:
455:
402:
370:
312:
289:
238:
199:
173:
143:
683:under these transformations differs from that of
2785:: CS1 maint: bot: original URL status unknown (
1302:{\displaystyle \sin t,\tan t,\cos(2t),\tan(t/2)}
46:but its sources remain unclear because it lacks
2765:Revue de mathématiques et de sciences physiques
897:{\displaystyle \int \sin ^{p}(t)\cos ^{q}(t)dt}
822:in a new variable, which can be calculated by
8:
2023:{\displaystyle \int -{\frac {du}{1-u^{2}}},}
671:Because rules 1 and 2 involve flipping the
320:, under translation and reflections of the
541:{\displaystyle \omega (\pi +t)=\omega (t)}
456:{\displaystyle \omega (\pi -t)=\omega (t)}
2702:
2668:
2648:
2632:
2598:
2595:
2590:
2539:
2531:
2502:
2487:
2464:
2449:
2436:
2422:
2393:
2373:
2342:
2336:
2281:
2276:
2199:
2183:
2154:
2149:
2106:
2085:
2070:
2058:
2042:
2040:
2008:
1987:
1979:
1936:
1912:
1891:
1876:
1847:
1845:
1808:
1791:{\displaystyle \int {\frac {dt}{\sin t}}}
1765:
1760:
1697:
1668:
1636:
1590:
1560:
1546:
1502:
1482:
1461:
1449:
1444:). In every case, the change of variable
1423:
1352:
1314:
1288:
1229:
1191:
1135:
1120:{\displaystyle \int g(\cosh t,\sinh t)dt}
1076:
1022:
973:
921:
870:
848:
839:
740:
723:
710:These rules can be, in fact, stated as a
646:
626:
588:
553:
503:
468:
418:
383:
336:
302:
280:
251:
229:
212:
186:
160:
127:
92:, formulated by the French mathematician
77:Learn how and when to remove this message
1067:Another version for hyperbolic functions
2747:
1179:{\displaystyle \int g(\cos t,\sin t)dt}
2778:
1130:If Bioche's rules suggest calculating
371:{\displaystyle \omega (-t)=\omega (t)}
703:) that has the same symmetry as
7:
290:{\displaystyle \omega (t)=f(t)\,dt}
2599:
1969:This transforms the integral into
14:
1573:{\displaystyle \int \sin t\,dt.}
904:, Bioche's rules apply as well.
679:, and therefore the behavior of
548:, a good change of variables is
463:, a good change of variables is
378:, a good change of variables is
23:
2357:{\displaystyle \beta ^{2}<1}
1536:As a trivial example, consider
2755:Vidiani, L.G. (October 1976).
2510:
2496:
2472:
2458:
2401:
2387:
1735:{\displaystyle -u+c=-\cos t+c}
1601:
1595:
1510:
1496:
1431:
1417:
1405:
1396:
1384:
1378:
1366:
1360:
1334:
1328:
1296:
1282:
1270:
1261:
1211:
1205:
1167:
1143:
1108:
1084:
1042:
1036:
993:
987:
944:
935:
885:
879:
863:
857:
824:partial fraction decomposition
802:
778:
770:
746:
734:
728:
654:
640:
535:
529:
520:
508:
450:
444:
435:
423:
365:
359:
350:
341:
277:
271:
262:
256:
226:
220:
138:
132:
1:
239:{\displaystyle \int f(t)\,dt}
1516:{\displaystyle u=\tanh(t/2)}
675:axis, they flip the sign of
2407:{\displaystyle u=\tan(t/2)}
1619:{\displaystyle f(t)=\sin t}
1071:Suppose one is calculating
691:, stating them in terms of
660:{\displaystyle u=\tan(t/2)}
327:Bioche's rules state that:
2844:
2526:and a second substitution
1340:{\displaystyle u=\cosh(t)}
1058:If not, one is reduced to
950:{\displaystyle u=\cos(2t)}
834:To calculate the integral
818:) to the integration of a
2759:[Bioche's rules]
1217:{\displaystyle u=\cos(t)}
1048:{\displaystyle u=\sin(t)}
999:{\displaystyle u=\cos(t)}
611:{\displaystyle u=\cos 2t}
246:, consider the integrand
2364:. Although the function
1828:{\displaystyle u=\cos t}
1685:{\displaystyle -\int du}
1656:{\displaystyle u=\cos t}
718:is actually of the form
621:In all other cases, use
573:{\displaystyle u=\tan t}
488:{\displaystyle u=\sin t}
403:{\displaystyle u=\cos t}
207:. In order to calculate
32:This article includes a
2800:Handbook of Integration
1470:{\displaystyle u=e^{t}}
61:more precise citations.
2730:
2574:
2520:
2408:
2358:
2320:
2250:
2139:. The result is that
2133:
2024:
1958:
1829:
1792:
1736:
1686:
1657:
1620:
1574:
1517:
1471:
1438:
1341:
1303:
1218:
1180:
1121:
1049:
1000:
951:
898:
812:
661:
612:
574:
542:
489:
457:
404:
372:
314:
291:
240:
201:
200:{\displaystyle \cos t}
175:
174:{\displaystyle \sin t}
145:
2731:
2575:
2521:
2409:
2359:
2321:
2251:
2134:
2025:
1959:
1830:
1793:
1737:
1687:
1658:
1621:
1575:
1518:
1472:
1439:
1342:
1304:
1219:
1181:
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1050:
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813:
662:
613:
575:
543:
490:
458:
405:
373:
315:
292:
241:
202:
176:
146:
2828:Theorems in calculus
2589:
2580:leads to the result
2530:
2421:
2372:
2335:
2275:
2148:
2039:
1978:
1844:
1807:
1759:
1696:
1667:
1635:
1589:
1545:
1481:
1448:
1351:
1313:
1228:
1190:
1134:
1075:
1021:
972:
920:
838:
722:
625:
587:
552:
502:
467:
417:
382:
335:
301:
250:
211:
185:
159:
144:{\displaystyle f(t)}
126:
105:indefinite integrals
2809:, pp. 190−197.
830:Case of polynomials
153:rational expression
2757:"Règles de Bioche"
2726:
2570:
2516:
2404:
2354:
2316:
2246:
2129:
2020:
1954:
1825:
1788:
1732:
1682:
1653:
1616:
1570:
1513:
1467:
1434:
1337:
1299:
1214:
1176:
1117:
1045:
996:
947:
916:are odd, one uses
894:
808:
657:
608:
570:
538:
485:
453:
400:
368:
310:
287:
236:
197:
171:
141:
122:In the following,
34:list of references
2823:Integral calculus
2710:
2694:
2693:
2655:
2654:
2627:
2565:
2564:
2514:
2311:
2235:
2191:
2175:
2122:
2101:
2078:
2065:
2015:
1949:
1925:
1904:
1868:
1786:
1750:The integrand in
820:rational function
806:
87:
86:
79:
2835:
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2784:
2776:
2774:
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2735:
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2063:
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2027:
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2021:
2016:
2014:
2013:
2012:
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1988:
1963:
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1941:
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1903:
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1797:
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1118:
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997:
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948:
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150:
148:
147:
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102:
82:
75:
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68:
62:
57:this article by
48:inline citations
27:
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19:
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1224:(respectively,
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968:even, one uses
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313:{\textstyle dt}
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38:related reading
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1065:
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1038:
1035:
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1029:
1026:
1017:odd, one uses
1007:
995:
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989:
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235:
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196:
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167:
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140:
137:
134:
131:
94:Charles Bioche
90:Bioche's rules
85:
84:
42:external links
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
2840:
2829:
2826:
2824:
2821:
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2818:
2808:
2804:
2802:, p. 108
2801:
2797:
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2017:
2009:
2005:
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1998:
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1984:
1981:
1974:
1973:
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1951:
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1234:
1231:
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1016:
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990:
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963:
959:
941:
938:
932:
929:
926:
923:
915:
911:
907:
906:
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888:
882:
876:
871:
867:
860:
854:
849:
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841:
829:
827:
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796:
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790:
787:
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781:
775:
767:
764:
761:
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749:
743:
737:
731:
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713:
708:
706:
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682:
678:
674:
651:
647:
643:
637:
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631:
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596:
593:
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582:
567:
564:
561:
558:
555:
532:
526:
523:
517:
514:
511:
505:
497:
482:
479:
476:
473:
470:
447:
441:
438:
432:
429:
426:
420:
412:
397:
394:
391:
388:
385:
362:
356:
353:
347:
344:
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323:
307:
304:
284:
281:
274:
268:
265:
259:
253:
233:
230:
223:
217:
214:
194:
191:
188:
168:
165:
162:
154:
135:
129:
120:
118:
114:
110:
107:in which the
106:
100:
95:
91:
81:
78:
70:
60:
56:
50:
49:
43:
39:
35:
30:
21:
20:
2806:
2799:
2798:Zwillinger,
2781:cite journal
2769:. Retrieved
2764:
2750:
2416:
2365:
2330:
2265:
2034:
1968:
1802:
1749:
1627:
1584:
1535:
1129:
1070:
1014:
1013:is even and
1010:
965:
961:
913:
909:
833:
715:
709:
704:
700:
696:
692:
688:
684:
680:
676:
672:
670:
326:
321:
121:
89:
88:
73:
64:
53:Please help
45:
964:is odd and
97: [
59:introducing
2817:Categories
2742:References
2805:Stewart,
2700:
2690:β
2679:β
2676:−
2661:
2646:β
2642:−
2621:
2615:β
2593:∫
2561:β
2550:β
2547:−
2494:
2456:
2443:−
2428:
2385:
2340:β
2305:
2299:β
2279:∫
2266:Consider
2262:Example 3
2229:
2223:−
2212:
2197:
2181:−
2169:
2152:∫
2116:−
2052:−
2002:−
1985:−
1982:∫
1943:
1930:−
1910:−
1898:
1874:−
1862:
1820:
1780:
1763:∫
1746:Example 2
1721:
1715:−
1700:−
1674:∫
1671:−
1648:
1611:
1555:
1549:∫
1532:Example 1
1494:
1415:
1394:
1376:
1358:
1326:
1280:
1259:
1247:
1235:
1203:
1162:
1150:
1138:∫
1103:
1091:
1079:∫
1034:
985:
933:
877:
855:
842:∫
797:
785:
765:
753:
638:
600:
565:
527:ω
512:π
506:ω
480:
442:ω
430:−
427:π
421:ω
395:
357:ω
345:−
339:ω
254:ω
215:∫
192:
166:
111:contains
109:integrand
67:June 2022
1527:Examples
1060:lineariz
2771:10 June
712:theorem
117:cosines
55:improve
2658:arctan
2417:Using
2331:where
1924:
2761:(PDF)
1835:. So
1585:Then
151:is a
113:sines
101:]
40:, or
2787:link
2773:2022
2349:<
1491:tanh
1412:tanh
1391:cosh
1373:tanh
1355:sinh
1323:cosh
1100:sinh
1088:cosh
912:and
181:and
115:and
2697:tan
2618:cos
2485:tan
2447:tan
2425:cos
2382:tan
2302:cos
2226:cos
2209:cos
2166:sin
1934:cos
1889:sin
1859:sin
1817:cos
1777:sin
1718:cos
1645:cos
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1552:sin
1523:).
1277:tan
1256:cos
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1232:sin
1200:cos
1186:by
1159:sin
1147:cos
1031:sin
1009:If
982:cos
960:If
930:cos
908:If
868:cos
846:sin
794:cos
782:sin
762:cos
750:sin
635:tan
597:cos
562:tan
498:If
477:sin
413:If
392:cos
331:If
189:cos
163:sin
155:in
2819::
2783:}}
2779:{{
2763:.
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2194:ln
1127:.
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