3011:
2474:
3006:{\displaystyle {\begin{alignedat}{4}\int 2\sin(x)\cos(x)\,dx=&&\sin ^{2}(x)+C=&&-\cos ^{2}(x)+1+C=&&-{\frac {1}{2}}\cos(2x)+{\frac {1}{2}}+C\\\int 2\sin(x)\cos(x)\,dx=&&-\cos ^{2}(x)+C=&&\sin ^{2}(x)-1+C=&&-{\frac {1}{2}}\cos(2x)-{\frac {1}{2}}+C\\\int 2\sin(x)\cos(x)\,dx=&&-{\frac {1}{2}}\cos(2x)+C=&&\sin ^{2}(x)+C=&&-\cos ^{2}(x)+C\\\end{alignedat}}}
25:
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1722:
1450:
2418:
However, different methods of computation of indefinite integrals can result in multiple resulting antiderivatives, each implicitly containing different constants of integration, and no particular option may be considered simplest. For example,
830:
2058:
It turns out that adding and subtracting constants is the only flexibility available in finding different antiderivatives of the same function. That is, all antiderivatives are the same up to a constant. To express this fact for
3228:
is the space of all constant functions. The process of indefinite integration amounts to finding a pre-image of a given function. There is no canonical pre-image for a given function, but the set of all such pre-images forms a
2222:
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1103:
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525:
3097:
3017:. A general solution containing the arbitrary constant is often necessary to identify the correct particular solution. For example, to obtain the antiderivative of
665:
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490:
466:
167:
147:
4486:
4371:
690:
4354:
4349:
4359:
4344:
3458:
3106:. Almost all differential equations will have many solutions, and each constant represents the unique solution of a well-posed initial value problem.
3646:
42:
4339:
3956:
3710:
3424:
3013:
Additionally, omission of the constant, or setting it to zero, may make it prohibitive to deal with a number of problems, such as those with
2393:{\displaystyle {\begin{aligned}{\frac {d}{dx}}&={\frac {d}{dx}}\sin(x)+{\frac {d}{dx}}C\\&=\cos(x)+0\\&=\cos(x)\end{aligned}}}
1493:
were 0, then it would not be possible to integrate from 0 to 3, because the function is not defined between 1 and 2. Here, there will be
1717:{\displaystyle \int {\frac {dx}{x}}={\begin{cases}\ln \left|x\right|+C^{-}&x<0\\\ln \left|x\right|+C^{+}&x>0\end{cases}}}
3508:
89:
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61:
3868:
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2407:
The inclusion of the constant of integration is necessitated in some, but not all circumstances. For instance, when evaluating
1266:
832:
The constant is a way of expressing that every function with at least one antiderivative will have an infinite number of them.
68:
4082:
3774:
46:
4059:
1445:{\displaystyle {\begin{aligned}&0=\int _{a}^{x}F'(t)\,dt\\&0=F(x)-F(a)\\&0=F(x)-C\\&F(x)=C\\\end{aligned}}}
4172:
4110:
3905:
3779:
3451:
3262:
75:
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3658:
3636:
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3846:
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1197:
making the goal to prove that an everywhere differentiable function whose derivative is always zero must be constant:
4232:
874:
838:
57:
4512:
3851:
3621:
3298:
35:
4270:
4217:
1506:
3678:
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4157:
3705:
3444:
1509:, one can extend this theorem to disconnected domains. For example, there are two constants of integration for
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1836:
280:
an additive constant. This constant expresses an ambiguity inherent in the construction of antiderivatives.
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3233:. Choosing a constant is the same as choosing an element of the coset. In this context, solving an
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3966:
3961:
3757:
3528:
1489:. For example, if one were to ask for functions defined on the union of intervals and , and if
1481:. If the real line were not connected, one would not always be able to integrate from our fixed
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3900:
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are not differentiable at even one point, then the theorem might fail. As an example, let
1502:
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1203:
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2009:
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a valid antiderivative). For that reason, the indefinite integral is often written as
4506:
4275:
4130:
4016:
3720:
3695:
3192:
maps a function to zero if and only if that function is constant. Consequently, the
3102:
The constant of integration also implicitly or explicitly appears in the language of
2187:. It is easily determined that all of the following functions are antiderivatives of
825:{\textstyle {\frac {d}{dx}}(F(x)+C)={\frac {d}{dx}}F(x)+{\frac {d}{dx}}C=F'(x)=f(x).}
273:
170:
4285:
4255:
4120:
3683:
3118:
2415:, the constant of integration can be ignored as it will always cancel with itself.
3409:
3533:
3475:
3306:
3274:
3114:
24:
4250:
4182:
3936:
3809:
3673:
3663:
3606:
3238:
603:
4444:
4192:
4187:
3498:
4439:
3941:
3819:
3467:
3411:
The calculus lifesaver : all the tools you need to excel at calculus
122:
606:
of any constant function is zero. Once one has found one antiderivative
4290:
3543:
4459:
3523:
590:
although the constant of integration might be sometimes omitted in
3538:
3230:
277:
3440:
18:
2006:
differentiable the theorem still fails. As an example, take
1477:
Two facts are crucial in this proof. First, the real line is
3436:
3113:. The space of all (suitable) real-valued functions on the
1710:
1579:, so for example, the general form for the integral of 1/
907:
be two everywhere differentiable functions. Suppose that
3331:"Definition of constant of integration | Dictionary.com"
1962:
do not differ by a constant, even if it is assumed that
16:
Constant expressing ambiguity from indefinite integrals
3202:
3166:
3130:
1549:
1515:
1269:, together with the assumption that the derivative of
693:
533:
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can be integrated in at least three different ways.
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4323:
4241:
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4075:
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3914:
3795:
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3552:
3474:
1902:is zero where it is defined, and the derivative of
49:. Unsourced material may be challenged and removed.
3415:. Princeton : Princeton University Press. p.
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2018:
1994:
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484:
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402:
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304:
264:
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161:
141:
1766:were assumed to be everywhere differentiable. If
900:{\displaystyle G:\mathbb {R} \to \mathbb {R} }
864:{\displaystyle F:\mathbb {R} \to \mathbb {R} }
3452:
687:will give us another antiderivative, because
8:
2156:{\displaystyle \int \cos(x)\,dx=\sin(x)+C,}
4327:
3459:
3445:
3437:
1505:. In general, by replacing constants with
4487:Regiomontanus' angle maximization problem
3203:
3201:
3167:
3165:
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3129:
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2011:
1987:
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992:
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549:
532:
497:
477:
453:
415:
386:
350:
321:
288:
248:
215:
182:
154:
134:
109:Learn how and when to remove this message
3835:Differentiating under the integral sign
3254:
3109:An additional justification comes from
1839:, which is zero for negative values of
468:is an arbitrary constant (meaning that
3711:Inverse functions and differentiation
7:
1922:is always zero. Yet it's clear that
47:adding citations to reliable sources
1843:and one for non-negative values of
667:adding or subtracting any constant
3509:Free variables and bound variables
967:. Then there exists a real number
583:{\textstyle \int f(x)\,dx=F(x)+C,}
14:
4314:The Method of Mechanical Theorems
283:More specifically, if a function
3869:Partial fractions in integration
3785:Stochastic differential equation
23:
4007:Jacobian matrix and determinant
3862:Tangent half-angle substitution
3830:Fundamental theorem of calculus
3269:Calculus: Early Transcendentals
3237:is interpreted as lying in the
2462:{\displaystyle 2\sin(x)\cos(x)}
2413:fundamental theorem of calculus
1267:fundamental theorem of calculus
956:{\displaystyle F\,'(x)=G\,'(x)}
34:needs additional citations for
4083:Arithmetico-geometric sequence
3775:Ordinary differential equation
3036:
3030:
2990:
2984:
2953:
2947:
2919:
2910:
2875:
2869:
2860:
2854:
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2767:
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2627:
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2581:
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2544:
2515:
2509:
2500:
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2456:
2450:
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2352:
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2300:
2269:
2260:
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2200:
2141:
2135:
2116:
2110:
2078:
2072:
2002:are everywhere continuous and
1863:
1857:
1822:
1816:
1429:
1423:
1406:
1400:
1383:
1377:
1368:
1362:
1338:
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1245:
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1018:
1012:
1003:
997:
950:
944:
929:
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889:
853:
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810:
801:
795:
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730:
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715:
709:
651:
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622:
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568:
562:
546:
540:
508:
502:
426:
420:
397:
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361:
355:
332:
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299:
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259:
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226:
220:
193:
187:
1:
3906:Integro-differential equation
3780:Partial differential equation
3301:; Edwards, Bruce H. (2009).
3221:{\textstyle {\frac {d}{dx}}}
3185:{\textstyle {\frac {d}{dx}}}
3149:{\textstyle {\frac {d}{dx}}}
3053:= π, then only one value of
1572:{\textstyle \int \tan x\,dx}
4060:Generalized Stokes' theorem
3847:Integration by substitution
1044:To prove this, notice that
1030:{\displaystyle F(x)-G(x)=C}
4531:
3589:(ε, δ)-definition of limit
3049:that has the value 400 at
1543:, and infinitely many for
1507:locally constant functions
410:is given by the functions
243:of all antiderivatives of
4482:Proof that 22/7 exceeds π
4397:
4271:Gottfried Wilhelm Leibniz
4218:e (mathematical constant)
3356:"Constant of Integration"
1289:vanishes, implying that
1174:by the constant function
58:"Constant of integration"
4233:Stirling's approximation
3706:Implicit differentiation
3654:Rules of differentiation
3073:will work (in this case
3015:initial value conditions
2087:{\displaystyle \cos(x),}
1497:constants, one for each
1474:is a constant function.
345:is an antiderivative of
4467:Euler–Maclaurin formula
4372:trigonometric functions
3825:Constant of integration
3407:Banner, Adrian (2007).
3042:{\displaystyle \cos(x)}
2212:{\displaystyle \cos(x)}
2185:constant of integration
1882:Then the derivative of
1875:{\displaystyle G(x)=0.}
1837:Heaviside step function
1254:{\displaystyle C=F(a).}
441:{\displaystyle F(x)+C,}
127:constant of integration
4436:Differential geometry
4281:Infinitesimal calculus
3984:Multivariable calculus
3932:Directional derivative
3738:Second derivative test
3716:Logarithmic derivative
3689:General Leibniz's rule
3584:Order of approximation
3222:
3186:
3150:
3104:differential equations
3093:
3067:
3043:
3007:
2463:
2394:
2213:
2177:
2157:
2088:
2050:
2020:
1996:
1976:
1956:
1936:
1916:
1896:
1876:
1829:
1800:
1780:
1760:
1740:
1718:
1573:
1537:
1536:{\textstyle \int dx/x}
1468:
1446:
1283:
1255:
1217:
1191:
1168:
1148:
1119:
1099:
1037:for every real number
1031:
981:
963:for every real number
957:
901:
865:
826:
681:
661:
629:
584:
521:
520:{\displaystyle F(x)+C}
486:
462:
442:
404:
371:
339:
306:
266:
233:
200:
163:
143:
4355:logarithmic functions
4350:exponential functions
4266:Generality of algebra
4144:Tests of convergence
3770:Differential equation
3754:Further applications
3743:Extreme value theorem
3733:First derivative test
3627:Differential operator
3599:Differential calculus
3360:mathworld.wolfram.com
3235:initial value problem
3223:
3187:
3151:
3123:differential operator
3094:
3092:{\displaystyle C=400}
3068:
3044:
3008:
2464:
2395:
2214:
2178:
2158:
2089:
2051:
2021:
1997:
1977:
1957:
1937:
1917:
1897:
1877:
1830:
1801:
1781:
1761:
1741:
1719:
1574:
1538:
1469:
1454:thereby showing that
1447:
1284:
1256:
1218:
1200:Choose a real number
1192:
1169:
1149:
1120:
1100:
1032:
982:
958:
902:
866:
827:
682:
662:
660:{\displaystyle f(x),}
630:
585:
522:
487:
463:
443:
405:
372:
370:{\displaystyle f(x),}
340:
307:
267:
234:
206:to indicate that the
201:
164:
144:
4420:Miscellaneous topics
4360:hyperbolic functions
4345:irrational functions
4223:Exponential function
4076:Sequences and series
3842:Integration by parts
3200:
3164:
3128:
3077:
3057:
3021:
2475:
2423:
2223:
2191:
2167:
2098:
2063:
2049:{\displaystyle G=0.}
2034:
2010:
1986:
1966:
1946:
1926:
1906:
1886:
1851:
1828:{\displaystyle F(x)}
1810:
1790:
1770:
1750:
1730:
1589:
1547:
1513:
1458:
1295:
1273:
1227:
1204:
1178:
1158:
1147:{\displaystyle F-G,}
1129:
1109:
1098:{\displaystyle '=0.}
1048:
991:
971:
911:
875:
839:
691:
671:
639:
628:{\displaystyle F(x)}
610:
531:
496:
476:
452:
414:
403:{\displaystyle f(x)}
385:
349:
338:{\displaystyle F(x)}
320:
305:{\displaystyle f(x)}
287:
265:{\displaystyle f(x)}
247:
232:{\displaystyle f(x)}
214:
199:{\displaystyle f(x)}
181:
153:
133:
43:improve this article
4407:List of derivatives
4243:History of calculus
4158:Cauchy condensation
4055:Exterior derivative
4012:Lagrange multiplier
3748:Maximum and minimum
3579:Limit of a sequence
3567:Limit of a function
3514:Graph of a function
3494:Continuous function
3380:Reader Survey: log|
3354:Weisstein, Eric W.
1499:connected component
1323:
1125:can be replaced by
381:antiderivatives of
208:indefinite integral
129:, often denoted by
4340:rational functions
4307:Method of Fluxions
4153:Alternating series
4050:Differential forms
4032:Partial derivative
3992:Divergence theorem
3874:Quadratic integral
3642:Leibniz's notation
3632:Mean value theorem
3617:Partial derivative
3562:Indeterminate form
3335:www.dictionary.com
3243:initial conditions
3218:
3182:
3146:
3089:
3063:
3039:
3003:
3001:
2459:
2409:definite integrals
2390:
2388:
2209:
2173:
2153:
2084:
2046:
2016:
1992:
1972:
1952:
1932:
1912:
1892:
1872:
1825:
1796:
1776:
1756:
1736:
1714:
1709:
1569:
1533:
1464:
1442:
1440:
1309:
1279:
1251:
1216:{\displaystyle a,}
1213:
1190:{\displaystyle 0,}
1187:
1164:
1144:
1115:
1095:
1027:
977:
953:
897:
861:
822:
677:
657:
625:
592:lists of integrals
580:
517:
482:
458:
438:
400:
367:
335:
302:
276:, is only defined
262:
229:
196:
159:
139:
4513:Integral calculus
4500:
4499:
4426:Complex calculus
4415:
4414:
4296:Law of Continuity
4228:Natural logarithm
4213:Bernoulli numbers
4204:Special functions
4163:Direct comparison
4027:Multiple integral
3901:Integral equation
3797:Integral calculus
3728:Stationary points
3702:Other techniques
3647:Newton's notation
3612:Second derivative
3504:Finite difference
3426:978-0-691-13088-0
3389:", Tom Leinster,
3216:
3180:
3144:
3066:{\displaystyle C}
2902:
2830:
2799:
2650:
2619:
2325:
2292:
2243:
2176:{\displaystyle C}
2019:{\displaystyle F}
2004:almost everywhere
1995:{\displaystyle G}
1975:{\displaystyle F}
1955:{\displaystyle G}
1935:{\displaystyle F}
1915:{\displaystyle G}
1895:{\displaystyle F}
1799:{\displaystyle G}
1779:{\displaystyle F}
1759:{\displaystyle G}
1739:{\displaystyle F}
1608:
1467:{\displaystyle F}
1282:{\displaystyle F}
1167:{\displaystyle G}
1118:{\displaystyle F}
980:{\displaystyle C}
779:
749:
707:
680:{\displaystyle C}
485:{\displaystyle C}
461:{\displaystyle C}
312:is defined on an
162:{\displaystyle c}
142:{\displaystyle C}
119:
118:
111:
93:
4520:
4430:Contour integral
4328:
4178:Limit comparison
4087:Types of series
4046:Advanced topics
4037:Surface integral
3881:Trapezoidal rule
3820:Basic properties
3815:Riemann integral
3763:Taylor's theorem
3489:Concave function
3484:Binomial theorem
3461:
3454:
3447:
3438:
3431:
3430:
3414:
3404:
3398:
3397:, March 19, 2012
3376:
3370:
3369:
3367:
3366:
3351:
3345:
3344:
3342:
3341:
3327:
3321:
3320:
3305:(9th ed.).
3295:
3289:
3288:
3273:(6th ed.).
3272:
3259:
3227:
3225:
3224:
3219:
3217:
3215:
3204:
3191:
3189:
3188:
3183:
3181:
3179:
3168:
3155:
3153:
3152:
3147:
3145:
3143:
3132:
3111:abstract algebra
3098:
3096:
3095:
3090:
3072:
3070:
3069:
3064:
3048:
3046:
3045:
3040:
3012:
3010:
3009:
3004:
3002:
2980:
2979:
2966:
2943:
2942:
2932:
2903:
2895:
2889:
2831:
2823:
2800:
2792:
2786:
2757:
2756:
2746:
2723:
2722:
2709:
2651:
2643:
2620:
2612:
2606:
2577:
2576:
2563:
2540:
2539:
2529:
2468:
2466:
2465:
2460:
2399:
2397:
2396:
2391:
2389:
2364:
2333:
2326:
2324:
2313:
2293:
2291:
2280:
2244:
2242:
2231:
2218:
2216:
2215:
2210:
2182:
2180:
2179:
2174:
2162:
2160:
2159:
2154:
2093:
2091:
2090:
2085:
2055:
2053:
2052:
2047:
2025:
2023:
2022:
2017:
2001:
1999:
1998:
1993:
1981:
1979:
1978:
1973:
1961:
1959:
1958:
1953:
1941:
1939:
1938:
1933:
1921:
1919:
1918:
1913:
1901:
1899:
1898:
1893:
1881:
1879:
1878:
1873:
1834:
1832:
1831:
1826:
1805:
1803:
1802:
1797:
1785:
1783:
1782:
1777:
1765:
1763:
1762:
1757:
1745:
1743:
1742:
1737:
1723:
1721:
1720:
1715:
1713:
1712:
1695:
1694:
1682:
1650:
1649:
1637:
1609:
1604:
1596:
1578:
1576:
1575:
1570:
1542:
1540:
1539:
1534:
1529:
1473:
1471:
1470:
1465:
1451:
1449:
1448:
1443:
1441:
1418:
1389:
1351:
1331:
1322:
1317:
1301:
1288:
1286:
1285:
1280:
1260:
1258:
1257:
1252:
1222:
1220:
1219:
1214:
1196:
1194:
1193:
1188:
1173:
1171:
1170:
1165:
1153:
1151:
1150:
1145:
1124:
1122:
1121:
1116:
1104:
1102:
1101:
1096:
1088:
1036:
1034:
1033:
1028:
986:
984:
983:
978:
962:
960:
959:
954:
943:
922:
906:
904:
903:
898:
896:
888:
870:
868:
867:
862:
860:
852:
831:
829:
828:
823:
794:
780:
778:
767:
750:
748:
737:
708:
706:
695:
686:
684:
683:
678:
666:
664:
663:
658:
634:
632:
631:
626:
594:for simplicity.
589:
587:
586:
581:
526:
524:
523:
518:
491:
489:
488:
483:
467:
465:
464:
459:
447:
445:
444:
439:
409:
407:
406:
401:
377:then the set of
376:
374:
373:
368:
344:
342:
341:
336:
311:
309:
308:
303:
274:connected domain
271:
269:
268:
263:
238:
236:
235:
230:
205:
203:
202:
197:
168:
166:
165:
160:
148:
146:
145:
140:
114:
107:
103:
100:
94:
92:
51:
27:
19:
4530:
4529:
4523:
4522:
4521:
4519:
4518:
4517:
4503:
4502:
4501:
4496:
4492:Steinmetz solid
4477:Integration Bee
4411:
4393:
4319:
4261:Colin Maclaurin
4237:
4205:
4199:
4071:
4065:Tensor calculus
4042:Volume integral
3978:
3953:Basic theorems
3916:Vector calculus
3910:
3791:
3758:Newton's method
3593:
3572:One-sided limit
3548:
3529:Rolle's theorem
3519:Linear function
3470:
3465:
3435:
3434:
3427:
3406:
3405:
3401:
3377:
3373:
3364:
3362:
3353:
3352:
3348:
3339:
3337:
3329:
3328:
3324:
3317:
3297:
3296:
3292:
3285:
3261:
3260:
3256:
3251:
3208:
3198:
3197:
3172:
3162:
3161:
3160:. The operator
3158:linear operator
3136:
3126:
3125:
3075:
3074:
3055:
3054:
3019:
3018:
3000:
2999:
2971:
2965:
2934:
2931:
2888:
2839:
2838:
2785:
2748:
2745:
2714:
2708:
2659:
2658:
2605:
2568:
2562:
2531:
2528:
2473:
2472:
2421:
2420:
2405:
2387:
2386:
2362:
2361:
2331:
2330:
2317:
2284:
2272:
2235:
2221:
2220:
2189:
2188:
2165:
2164:
2096:
2095:
2094:one can write:
2061:
2060:
2032:
2031:
2028:Cantor function
2008:
2007:
1984:
1983:
1964:
1963:
1944:
1943:
1924:
1923:
1904:
1903:
1884:
1883:
1849:
1848:
1808:
1807:
1788:
1787:
1768:
1767:
1748:
1747:
1728:
1727:
1708:
1707:
1696:
1686:
1672:
1663:
1662:
1651:
1641:
1627:
1614:
1597:
1587:
1586:
1545:
1544:
1511:
1510:
1456:
1455:
1439:
1438:
1416:
1415:
1387:
1386:
1349:
1348:
1324:
1293:
1292:
1271:
1270:
1225:
1224:
1202:
1201:
1176:
1175:
1156:
1155:
1127:
1126:
1107:
1106:
1081:
1046:
1045:
989:
988:
969:
968:
938:
917:
909:
908:
873:
872:
837:
836:
787:
771:
741:
699:
689:
688:
669:
668:
637:
636:
635:for a function
608:
607:
600:
529:
528:
494:
493:
474:
473:
450:
449:
412:
411:
383:
382:
347:
346:
318:
317:
285:
284:
245:
244:
212:
211:
179:
178:
151:
150:
131:
130:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
4528:
4527:
4524:
4516:
4515:
4505:
4504:
4498:
4497:
4495:
4494:
4489:
4484:
4479:
4474:
4472:Gabriel's horn
4469:
4464:
4463:
4462:
4457:
4452:
4447:
4442:
4434:
4433:
4432:
4423:
4421:
4417:
4416:
4413:
4412:
4410:
4409:
4404:
4402:List of limits
4398:
4395:
4394:
4392:
4391:
4390:
4389:
4384:
4379:
4369:
4368:
4367:
4357:
4352:
4347:
4342:
4336:
4334:
4325:
4321:
4320:
4318:
4317:
4310:
4303:
4301:Leonhard Euler
4298:
4293:
4288:
4283:
4278:
4273:
4268:
4263:
4258:
4253:
4247:
4245:
4239:
4238:
4236:
4235:
4230:
4225:
4220:
4215:
4209:
4207:
4201:
4200:
4198:
4197:
4196:
4195:
4190:
4185:
4180:
4175:
4170:
4165:
4160:
4155:
4150:
4142:
4141:
4140:
4135:
4134:
4133:
4128:
4118:
4113:
4108:
4103:
4098:
4093:
4085:
4079:
4077:
4073:
4072:
4070:
4069:
4068:
4067:
4062:
4057:
4052:
4044:
4039:
4034:
4029:
4024:
4019:
4014:
4009:
4004:
4002:Hessian matrix
3999:
3994:
3988:
3986:
3980:
3979:
3977:
3976:
3975:
3974:
3969:
3964:
3959:
3957:Line integrals
3951:
3950:
3949:
3944:
3939:
3934:
3929:
3920:
3918:
3912:
3911:
3909:
3908:
3903:
3898:
3897:
3896:
3891:
3883:
3878:
3877:
3876:
3866:
3865:
3864:
3859:
3854:
3844:
3839:
3838:
3837:
3827:
3822:
3817:
3812:
3807:
3805:Antiderivative
3801:
3799:
3793:
3792:
3790:
3789:
3788:
3787:
3782:
3777:
3767:
3766:
3765:
3760:
3752:
3751:
3750:
3745:
3740:
3735:
3725:
3724:
3723:
3718:
3713:
3708:
3700:
3699:
3698:
3693:
3692:
3691:
3681:
3676:
3671:
3666:
3661:
3651:
3650:
3649:
3644:
3634:
3629:
3624:
3619:
3614:
3609:
3603:
3601:
3595:
3594:
3592:
3591:
3586:
3581:
3576:
3575:
3574:
3564:
3558:
3556:
3550:
3549:
3547:
3546:
3541:
3536:
3531:
3526:
3521:
3516:
3511:
3506:
3501:
3496:
3491:
3486:
3480:
3478:
3472:
3471:
3466:
3464:
3463:
3456:
3449:
3441:
3433:
3432:
3425:
3399:
3395:-category Café
3371:
3346:
3322:
3315:
3290:
3283:
3263:Stewart, James
3253:
3252:
3250:
3247:
3214:
3211:
3207:
3178:
3175:
3171:
3142:
3139:
3135:
3088:
3085:
3082:
3062:
3038:
3035:
3032:
3029:
3026:
2998:
2995:
2992:
2989:
2986:
2983:
2978:
2974:
2970:
2967:
2964:
2961:
2958:
2955:
2952:
2949:
2946:
2941:
2937:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2901:
2898:
2893:
2890:
2887:
2884:
2881:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2840:
2837:
2834:
2829:
2826:
2821:
2818:
2815:
2812:
2809:
2806:
2803:
2798:
2795:
2790:
2787:
2784:
2781:
2778:
2775:
2772:
2769:
2766:
2763:
2760:
2755:
2751:
2747:
2744:
2741:
2738:
2735:
2732:
2729:
2726:
2721:
2717:
2713:
2710:
2707:
2704:
2701:
2697:
2694:
2691:
2688:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2660:
2657:
2654:
2649:
2646:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2618:
2615:
2610:
2607:
2604:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2575:
2571:
2567:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2538:
2534:
2530:
2527:
2524:
2521:
2517:
2514:
2511:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2480:
2458:
2455:
2452:
2449:
2446:
2443:
2440:
2437:
2434:
2431:
2428:
2404:
2401:
2385:
2382:
2379:
2376:
2373:
2370:
2367:
2365:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2334:
2332:
2329:
2323:
2320:
2316:
2311:
2308:
2305:
2302:
2299:
2296:
2290:
2287:
2283:
2278:
2275:
2273:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2241:
2238:
2234:
2229:
2228:
2208:
2205:
2202:
2199:
2196:
2172:
2152:
2149:
2146:
2143:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2118:
2115:
2112:
2109:
2106:
2103:
2083:
2080:
2077:
2074:
2071:
2068:
2045:
2042:
2039:
2030:and again let
2015:
1991:
1971:
1951:
1931:
1911:
1891:
1871:
1868:
1865:
1862:
1859:
1856:
1824:
1821:
1818:
1815:
1795:
1775:
1755:
1735:
1711:
1706:
1703:
1700:
1697:
1693:
1689:
1685:
1681:
1678:
1675:
1671:
1668:
1665:
1664:
1661:
1658:
1655:
1652:
1648:
1644:
1640:
1636:
1633:
1630:
1626:
1623:
1620:
1619:
1617:
1612:
1607:
1603:
1600:
1594:
1568:
1565:
1561:
1558:
1555:
1552:
1532:
1528:
1524:
1521:
1518:
1463:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1367:
1364:
1361:
1358:
1355:
1352:
1350:
1347:
1344:
1340:
1337:
1334:
1330:
1327:
1321:
1316:
1312:
1308:
1305:
1302:
1300:
1278:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1212:
1209:
1186:
1183:
1163:
1143:
1140:
1137:
1134:
1114:
1094:
1091:
1087:
1084:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
976:
952:
949:
946:
942:
937:
934:
931:
928:
925:
921:
916:
895:
891:
887:
883:
880:
859:
855:
851:
847:
844:
821:
818:
815:
812:
809:
806:
803:
800:
797:
793:
790:
786:
783:
777:
774:
770:
765:
762:
759:
756:
753:
747:
744:
740:
735:
732:
729:
726:
723:
720:
717:
714:
711:
705:
702:
698:
676:
656:
653:
650:
647:
644:
624:
621:
618:
615:
599:
596:
579:
576:
573:
570:
567:
564:
561:
558:
555:
552:
548:
545:
542:
539:
536:
516:
513:
510:
507:
504:
501:
481:
457:
437:
434:
431:
428:
425:
422:
419:
399:
396:
393:
390:
366:
363:
360:
357:
354:
334:
331:
328:
325:
301:
298:
295:
292:
261:
258:
255:
252:
228:
225:
222:
219:
195:
192:
189:
186:
177:of a function
175:antiderivative
158:
138:
117:
116:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
4526:
4525:
4514:
4511:
4510:
4508:
4493:
4490:
4488:
4485:
4483:
4480:
4478:
4475:
4473:
4470:
4468:
4465:
4461:
4458:
4456:
4453:
4451:
4448:
4446:
4443:
4441:
4438:
4437:
4435:
4431:
4428:
4427:
4425:
4424:
4422:
4418:
4408:
4405:
4403:
4400:
4399:
4396:
4388:
4385:
4383:
4380:
4378:
4375:
4374:
4373:
4370:
4366:
4363:
4362:
4361:
4358:
4356:
4353:
4351:
4348:
4346:
4343:
4341:
4338:
4337:
4335:
4333:
4329:
4326:
4322:
4316:
4315:
4311:
4309:
4308:
4304:
4302:
4299:
4297:
4294:
4292:
4289:
4287:
4284:
4282:
4279:
4277:
4276:Infinitesimal
4274:
4272:
4269:
4267:
4264:
4262:
4259:
4257:
4254:
4252:
4249:
4248:
4246:
4244:
4240:
4234:
4231:
4229:
4226:
4224:
4221:
4219:
4216:
4214:
4211:
4210:
4208:
4202:
4194:
4191:
4189:
4186:
4184:
4181:
4179:
4176:
4174:
4171:
4169:
4166:
4164:
4161:
4159:
4156:
4154:
4151:
4149:
4146:
4145:
4143:
4139:
4136:
4132:
4129:
4127:
4124:
4123:
4122:
4119:
4117:
4114:
4112:
4109:
4107:
4104:
4102:
4099:
4097:
4094:
4092:
4089:
4088:
4086:
4084:
4081:
4080:
4078:
4074:
4066:
4063:
4061:
4058:
4056:
4053:
4051:
4048:
4047:
4045:
4043:
4040:
4038:
4035:
4033:
4030:
4028:
4025:
4023:
4020:
4018:
4017:Line integral
4015:
4013:
4010:
4008:
4005:
4003:
4000:
3998:
3995:
3993:
3990:
3989:
3987:
3985:
3981:
3973:
3970:
3968:
3965:
3963:
3960:
3958:
3955:
3954:
3952:
3948:
3945:
3943:
3940:
3938:
3935:
3933:
3930:
3928:
3925:
3924:
3922:
3921:
3919:
3917:
3913:
3907:
3904:
3902:
3899:
3895:
3892:
3890:
3889:Washer method
3887:
3886:
3884:
3882:
3879:
3875:
3872:
3871:
3870:
3867:
3863:
3860:
3858:
3855:
3853:
3852:trigonometric
3850:
3849:
3848:
3845:
3843:
3840:
3836:
3833:
3832:
3831:
3828:
3826:
3823:
3821:
3818:
3816:
3813:
3811:
3808:
3806:
3803:
3802:
3800:
3798:
3794:
3786:
3783:
3781:
3778:
3776:
3773:
3772:
3771:
3768:
3764:
3761:
3759:
3756:
3755:
3753:
3749:
3746:
3744:
3741:
3739:
3736:
3734:
3731:
3730:
3729:
3726:
3722:
3721:Related rates
3719:
3717:
3714:
3712:
3709:
3707:
3704:
3703:
3701:
3697:
3694:
3690:
3687:
3686:
3685:
3682:
3680:
3677:
3675:
3672:
3670:
3667:
3665:
3662:
3660:
3657:
3656:
3655:
3652:
3648:
3645:
3643:
3640:
3639:
3638:
3635:
3633:
3630:
3628:
3625:
3623:
3620:
3618:
3615:
3613:
3610:
3608:
3605:
3604:
3602:
3600:
3596:
3590:
3587:
3585:
3582:
3580:
3577:
3573:
3570:
3569:
3568:
3565:
3563:
3560:
3559:
3557:
3555:
3551:
3545:
3542:
3540:
3537:
3535:
3532:
3530:
3527:
3525:
3522:
3520:
3517:
3515:
3512:
3510:
3507:
3505:
3502:
3500:
3497:
3495:
3492:
3490:
3487:
3485:
3482:
3481:
3479:
3477:
3473:
3469:
3462:
3457:
3455:
3450:
3448:
3443:
3442:
3439:
3428:
3422:
3418:
3413:
3412:
3403:
3400:
3396:
3392:
3388:
3387:
3383:
3375:
3372:
3361:
3357:
3350:
3347:
3336:
3332:
3326:
3323:
3318:
3316:0-547-16702-4
3312:
3308:
3304:
3300:
3294:
3291:
3286:
3284:0-495-01166-5
3280:
3276:
3271:
3270:
3264:
3258:
3255:
3248:
3246:
3244:
3241:given by the
3240:
3236:
3232:
3212:
3209:
3205:
3195:
3176:
3173:
3169:
3159:
3140:
3137:
3133:
3124:
3120:
3116:
3112:
3107:
3105:
3100:
3086:
3083:
3080:
3060:
3052:
3033:
3027:
3024:
3016:
2996:
2993:
2987:
2981:
2976:
2972:
2968:
2962:
2959:
2956:
2950:
2944:
2939:
2935:
2928:
2925:
2922:
2916:
2913:
2907:
2904:
2899:
2896:
2891:
2885:
2882:
2879:
2872:
2866:
2863:
2857:
2851:
2848:
2845:
2842:
2835:
2832:
2827:
2824:
2819:
2813:
2810:
2804:
2801:
2796:
2793:
2788:
2782:
2779:
2776:
2773:
2770:
2764:
2758:
2753:
2749:
2742:
2739:
2736:
2730:
2724:
2719:
2715:
2711:
2705:
2702:
2699:
2692:
2686:
2683:
2677:
2671:
2668:
2665:
2662:
2655:
2652:
2647:
2644:
2639:
2633:
2630:
2624:
2621:
2616:
2613:
2608:
2602:
2599:
2596:
2593:
2590:
2584:
2578:
2573:
2569:
2565:
2559:
2556:
2553:
2547:
2541:
2536:
2532:
2525:
2522:
2519:
2512:
2506:
2503:
2497:
2491:
2488:
2485:
2482:
2470:
2453:
2447:
2444:
2438:
2432:
2429:
2426:
2416:
2414:
2410:
2402:
2400:
2380:
2374:
2371:
2368:
2366:
2358:
2355:
2349:
2343:
2340:
2337:
2335:
2327:
2321:
2318:
2314:
2309:
2303:
2297:
2294:
2288:
2285:
2281:
2276:
2274:
2266:
2263:
2257:
2251:
2248:
2239:
2236:
2232:
2203:
2197:
2194:
2186:
2170:
2150:
2147:
2144:
2138:
2132:
2129:
2126:
2123:
2120:
2113:
2107:
2104:
2101:
2081:
2075:
2069:
2066:
2056:
2043:
2040:
2037:
2029:
2013:
2005:
1989:
1969:
1949:
1929:
1909:
1889:
1869:
1866:
1860:
1854:
1846:
1842:
1838:
1819:
1813:
1793:
1773:
1753:
1733:
1724:
1704:
1701:
1698:
1691:
1687:
1683:
1679:
1676:
1673:
1669:
1666:
1659:
1656:
1653:
1646:
1642:
1638:
1634:
1631:
1628:
1624:
1621:
1615:
1610:
1605:
1601:
1598:
1592:
1584:
1582:
1566:
1563:
1559:
1556:
1553:
1550:
1530:
1526:
1522:
1519:
1516:
1508:
1504:
1500:
1496:
1492:
1488:
1485:to any given
1484:
1480:
1475:
1461:
1452:
1435:
1432:
1426:
1420:
1412:
1409:
1403:
1397:
1394:
1391:
1380:
1374:
1371:
1365:
1359:
1356:
1353:
1345:
1342:
1335:
1328:
1325:
1319:
1314:
1310:
1306:
1303:
1290:
1276:
1268:
1264:
1248:
1242:
1236:
1233:
1230:
1210:
1207:
1198:
1184:
1181:
1161:
1141:
1138:
1135:
1132:
1112:
1092:
1089:
1085:
1075:
1069:
1066:
1060:
1054:
1042:
1040:
1024:
1021:
1015:
1009:
1006:
1000:
994:
974:
966:
947:
940:
935:
932:
926:
919:
914:
881:
878:
845:
842:
833:
819:
813:
807:
804:
798:
791:
788:
784:
781:
775:
772:
768:
763:
757:
751:
745:
742:
738:
733:
727:
724:
718:
712:
703:
700:
696:
674:
654:
648:
642:
619:
613:
605:
597:
595:
593:
577:
574:
571:
565:
559:
556:
553:
550:
543:
537:
534:
514:
511:
505:
499:
479:
471:
455:
435:
432:
429:
423:
417:
394:
388:
380:
364:
358:
352:
329:
323:
315:
296:
290:
281:
279:
275:
256:
250:
242:
223:
217:
209:
190:
184:
176:
172:
171:constant term
156:
136:
128:
124:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
4387:Secant cubed
4312:
4305:
4286:Isaac Newton
4256:Brook Taylor
3923:Derivatives
3894:Shell method
3824:
3622:Differential
3410:
3402:
3394:
3390:
3385:
3381:
3374:
3363:. Retrieved
3359:
3349:
3338:. Retrieved
3334:
3325:
3302:
3293:
3268:
3257:
3119:vector space
3115:real numbers
3108:
3101:
3050:
2471:
2417:
2406:
2403:Significance
2184:
2057:
1844:
1840:
1725:
1585:
1580:
1494:
1490:
1486:
1482:
1476:
1453:
1291:
1262:
1199:
1043:
1038:
964:
834:
601:
469:
378:
282:
173:added to an
126:
120:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
4455:of surfaces
4206:and numbers
4168:Dirichlet's
4138:Telescoping
4091:Alternating
3679:L'HĂ´pital's
3476:Precalculus
3307:Brooks/Cole
3299:Larson, Ron
3275:Brooks/Cole
492:would make
239:(i.e., the
99:August 2020
4251:Adequality
3937:Divergence
3810:Arc length
3607:Derivative
3365:2020-08-14
3340:2020-08-14
3249:References
3239:hyperplane
3121:, and the
2411:using the
2026:to be the
1847:, and let
987:such that
604:derivative
69:newspapers
4450:of curves
4445:Curvature
4332:Integrals
4126:Maclaurin
4106:Geometric
3997:Geometric
3947:Laplacian
3659:linearity
3499:Factorial
3028:
2982:
2969:−
2945:
2908:
2892:−
2867:
2852:
2843:∫
2820:−
2805:
2789:−
2771:−
2759:
2725:
2712:−
2687:
2672:
2663:∫
2625:
2609:−
2579:
2566:−
2542:
2507:
2492:
2483:∫
2448:
2433:
2375:
2344:
2298:
2252:
2198:
2133:
2108:
2102:∫
2070:
1670:
1647:−
1625:
1593:∫
1557:
1551:∫
1517:∫
1479:connected
1410:−
1372:−
1311:∫
1136:−
1067:−
1007:−
890:→
854:→
535:∫
472:value of
4507:Category
4440:Manifold
4173:Integral
4116:Infinite
4111:Harmonic
4096:Binomial
3942:Gradient
3885:Volumes
3696:Quotient
3637:Notation
3468:Calculus
3303:Calculus
3265:(2008).
1726:Second,
1329:′
1261:For any
1223:and let
1086:′
941:′
920:′
792:′
314:interval
272:), on a
169:), is a
123:calculus
4377:inverse
4365:inverse
4291:Fluxion
4101:Fourier
3967:Stokes'
3962:Green's
3684:Product
3544:Tangent
1835:be the
1501:of the
83:scholar
4460:Tensor
4382:Secant
4148:Abel's
4131:Taylor
4022:Matrix
3972:Gauss'
3554:Limits
3534:Secant
3524:Radian
3423:
3313:
3281:
3194:kernel
2163:where
1503:domain
1265:, the
598:Origin
448:where
316:, and
125:, the
85:
78:
71:
64:
56:
4324:Lists
4183:Ratio
4121:Power
3857:Euler
3674:Chain
3664:Power
3539:Slope
3231:coset
3156:is a
3117:is a
278:up to
90:JSTOR
76:books
4193:Term
4188:Root
3927:Curl
3421:ISBN
3391:The
3384:| +
3311:ISBN
3279:ISBN
1982:and
1942:and
1786:and
1746:and
1702:>
1657:<
1583:is:
1154:and
871:and
835:Let
602:The
149:(or
62:news
3669:Sum
3417:380
3196:of
3099:).
3087:400
3025:cos
2973:cos
2936:sin
2905:cos
2864:cos
2849:sin
2802:cos
2750:sin
2716:cos
2684:cos
2669:sin
2622:cos
2570:cos
2533:sin
2504:cos
2489:sin
2445:cos
2430:sin
2372:cos
2341:cos
2295:sin
2249:sin
2195:cos
2183:is
2130:sin
2105:cos
2067:cos
1554:tan
1495:two
1105:So
470:any
379:all
241:set
210:of
121:In
45:by
4509::
3419:.
3358:.
3333:.
3309:.
3277:.
3245:.
2219::
2044:0.
1870:0.
1667:ln
1622:ln
1093:0.
1041:.
3460:e
3453:t
3446:v
3429:.
3393:n
3386:C
3382:x
3378:"
3368:.
3343:.
3319:.
3287:.
3213:x
3210:d
3206:d
3177:x
3174:d
3170:d
3141:x
3138:d
3134:d
3084:=
3081:C
3061:C
3051:x
3037:)
3034:x
3031:(
2997:C
2994:+
2991:)
2988:x
2985:(
2977:2
2963:=
2960:C
2957:+
2954:)
2951:x
2948:(
2940:2
2929:=
2926:C
2923:+
2920:)
2917:x
2914:2
2911:(
2900:2
2897:1
2886:=
2883:x
2880:d
2876:)
2873:x
2870:(
2861:)
2858:x
2855:(
2846:2
2836:C
2833:+
2828:2
2825:1
2817:)
2814:x
2811:2
2808:(
2797:2
2794:1
2783:=
2780:C
2777:+
2774:1
2768:)
2765:x
2762:(
2754:2
2743:=
2740:C
2737:+
2734:)
2731:x
2728:(
2720:2
2706:=
2703:x
2700:d
2696:)
2693:x
2690:(
2681:)
2678:x
2675:(
2666:2
2656:C
2653:+
2648:2
2645:1
2640:+
2637:)
2634:x
2631:2
2628:(
2617:2
2614:1
2603:=
2600:C
2597:+
2594:1
2591:+
2588:)
2585:x
2582:(
2574:2
2560:=
2557:C
2554:+
2551:)
2548:x
2545:(
2537:2
2526:=
2523:x
2520:d
2516:)
2513:x
2510:(
2501:)
2498:x
2495:(
2486:2
2457:)
2454:x
2451:(
2442:)
2439:x
2436:(
2427:2
2384:)
2381:x
2378:(
2369:=
2359:0
2356:+
2353:)
2350:x
2347:(
2338:=
2328:C
2322:x
2319:d
2315:d
2310:+
2307:)
2304:x
2301:(
2289:x
2286:d
2282:d
2277:=
2270:]
2267:C
2264:+
2261:)
2258:x
2255:(
2246:[
2240:x
2237:d
2233:d
2207:)
2204:x
2201:(
2171:C
2151:,
2148:C
2145:+
2142:)
2139:x
2136:(
2127:=
2124:x
2121:d
2117:)
2114:x
2111:(
2082:,
2079:)
2076:x
2073:(
2041:=
2038:G
2014:F
1990:G
1970:F
1950:G
1930:F
1910:G
1890:F
1867:=
1864:)
1861:x
1858:(
1855:G
1845:x
1841:x
1823:)
1820:x
1817:(
1814:F
1794:G
1774:F
1754:G
1734:F
1705:0
1699:x
1692:+
1688:C
1684:+
1680:|
1677:x
1674:|
1660:0
1654:x
1643:C
1639:+
1635:|
1632:x
1629:|
1616:{
1611:=
1606:x
1602:x
1599:d
1581:x
1567:x
1564:d
1560:x
1531:x
1527:/
1523:x
1520:d
1491:a
1487:x
1483:a
1462:F
1436:C
1433:=
1430:)
1427:x
1424:(
1421:F
1413:C
1407:)
1404:x
1401:(
1398:F
1395:=
1392:0
1384:)
1381:a
1378:(
1375:F
1369:)
1366:x
1363:(
1360:F
1357:=
1354:0
1346:t
1343:d
1339:)
1336:t
1333:(
1326:F
1320:x
1315:a
1307:=
1304:0
1277:F
1263:x
1249:.
1246:)
1243:a
1240:(
1237:F
1234:=
1231:C
1211:,
1208:a
1185:,
1182:0
1162:G
1142:,
1139:G
1133:F
1113:F
1090:=
1083:]
1079:)
1076:x
1073:(
1070:G
1064:)
1061:x
1058:(
1055:F
1052:[
1039:x
1025:C
1022:=
1019:)
1016:x
1013:(
1010:G
1004:)
1001:x
998:(
995:F
975:C
965:x
951:)
948:x
945:(
936:G
933:=
930:)
927:x
924:(
915:F
894:R
886:R
882::
879:G
858:R
850:R
846::
843:F
820:.
817:)
814:x
811:(
808:f
805:=
802:)
799:x
796:(
789:F
785:=
782:C
776:x
773:d
769:d
764:+
761:)
758:x
755:(
752:F
746:x
743:d
739:d
734:=
731:)
728:C
725:+
722:)
719:x
716:(
713:F
710:(
704:x
701:d
697:d
675:C
655:,
652:)
649:x
646:(
643:f
623:)
620:x
617:(
614:F
578:,
575:C
572:+
569:)
566:x
563:(
560:F
557:=
554:x
551:d
547:)
544:x
541:(
538:f
515:C
512:+
509:)
506:x
503:(
500:F
480:C
456:C
436:,
433:C
430:+
427:)
424:x
421:(
418:F
398:)
395:x
392:(
389:f
365:,
362:)
359:x
356:(
353:f
333:)
330:x
327:(
324:F
300:)
297:x
294:(
291:f
260:)
257:x
254:(
251:f
227:)
224:x
221:(
218:f
194:)
191:x
188:(
185:f
157:c
137:C
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
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