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Indefinite Integral

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An indefinite integral is an integral of the form

intf(z)dz,
(1)

i.e., without upper and lower limits, also called an antiderivative. The first fundamental theorem of calculus allows definite integrals to be computed in terms of indefinite integrals. In particular, this theorem states that if F is the indefinite integral for a complex function f(z), then

int_a^bf(z)dz=F(b)-F(a).
(2)

This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral. Indefinite integration is implemented in the Wolfram Language as Integrate[f, z].

Since the derivative of a constant is zero, any constant may be added to an antiderivative and will still correspond to the same integral. Another way of stating this is that the antiderivative is a nonunique inverse of the derivative. For this reason, indefinite integrals are often written in the form

intf(z)dz=F(z)+C,
(3)

where C is an arbitrary constant known as the constant of integration. The Wolfram Language returns indefinite integrals without explicit constants of integration. This means that, depending on the form used for the integrand, antiderivatives F_1 and F_2 can be obtained that differ by a constant (or, more generally, a piecewise constant function). It also means that Integrate[f+g, z] may differ from Integrate[f, z] + Integrate[g, z] by an arbitrary (piecewise) constant.

Note that indefinite integrals defined algebraically deal with complex quantities. However, many elementary calculus textbooks write formulas such as

int(dx)/x=ln|x|
(4)

(where the notation x is used to indicate that x is assumed to be a real number) instead of the complex variable version

int(dz)/z=lnz,
(5)

where z is generically a complex number (but also holds for real z). Defining a sort of "real-only" indefinite integral is perhaps done so that students can apply the first fundamental theorem of calculus using a Riemann integral and get correct answers while completely avoiding the use of complex analysis, multivalued functions, etc. (Although it should be noted that the first fundamental theorem of calculus only applies if the integrand is continuous on the interval of integration, so the additional stipulation must be made that int_a^bdx/x=[ln|x|]_b^a can be applied only if the interval [a,b] does not contain 0.)

However, this work (and the Wolfram Language) eschew the "real-only" definition, since inclusion of the absolute value means that the indefinite integral is no longer valid for a generic complex variable z (the presence of the |z| means the Cauchy-Riemann equations no longer can hold), and also violates the purely algebraic definition of indefinite integrals. Since physical problem involve definite integrals, it is much more sensible to stick with the usual complex/algebraic definitions of indefinite integration. In other words, while the Riemann integral

int_(-2)^(-1)(dx)/x=[ln|x|]_(-2)^(-1)=0-ln2=-ln2
(6)

gives the correct answer (and avoids complex quantities along the way), so does the complex integral

int_(-2)^(-1)(dz)/z=[lnz]_(-2)^(-1)=(ipi)-(ipi+ln2)=-ln2,
(7)

whereas the latter form preserves the benefits of genericness and at the same time prepares students for the extremely powerful tool of complex analysis which they should know about and will probably be learning about shortly in any case.

Liouville showed that the integrals

inte^(-x^2)dx, int(e^(-x))/xdx, int(sinx)/xdx, int(cosx)/xdx, int(dx)/(lnx)
(8)

cannot be expressed in terms of a finite number of elementary functions. These give rise to the functions

erf(x)=2/(sqrt(pi))inte^(-x^2)dx
(9)
Ei(x)=int(e^x)/xdx
(10)
Si(x)=int(sinx)/xdx
(11)
Ci(x)=int(cosx)/xdx
(12)
li(x)=int(dx)/(lnx)
(13)

(Havil 2003, p. 105), which are called erf, the exponential integral, sine integral, cosine integral, and logarithmic integral, respectively. The integral of any function of the form R(x)e^x, where R(x) is a rational function, reduces to elementary integrals and the function Ei(x) (Havil 2003, p. 106).

Other irreducibles include

intx^xdx, intx^(-x)dx, intsqrt(sinx)dx, intsqrt(cosx)dx, intsqrt(lnx)dx, inte^(e^x)dx, intlnlnxdx,
(14)

(cf. Marchisotto and Zakeri 1994), the last few of which can be written in closed form as

intsqrt(sinx)dx=-2E(1/4(pi-2x),sqrt(2))
(15)
intsqrt(cosx)dx=2E(1/2x,sqrt(2))
(16)
intsqrt(lnx)dx=-1/2sqrt(pi)erfi(sqrt(lnx))+xsqrt(lnx)
(17)
inte^(e^x)dx=Ei(e^x)
(18)
intln(lnx)dx=xlnlnx-li(x),
(19)

where E(x,k) is an elliptic integral of the second kind, erfi(x) is the erfi function, and Ei(x) is the exponential integral.

Chebyshev proved that if U, V, and W are rational numbers, then

intx^U(A+Bx^V)^Wdx
(20)

is integrable in terms of elementary functions iff (U+1)/V, W, or W+(U+1)/V is an integer (Ritt 1948, Shanks 1993).

Integration for general input is a tricky problem for symbolic mathematics software. In fact, many simple indefinite integrals, such as

int[d/(dz)Li_2(zlnz)]dz =-int[((lnz+1)ln(1-zlnz))/(zlnz)]dz int[d/(dz)(1/2sqrt(pi)erf(az)erf(bz))]dz =int[be^(-b^2z^2)erf(az)+ae^(-a^2z^2)erf(bz)]dz,
(21)

where Li_2(z) is the dilogarithm, cannot be done by very sophisticated software systems, including even the Wolfram Language.

More recently, Desmond (2026) used exhaustive symbolic integration to find elementary antiderivatives missed by several computer algebra systems. For example, direct differentiation verifies

int((2x+1)e^x+1)/(2sqrt(x+e^(-x)))dx=e^xsqrt(x+e^(-x))+C,
(22)

but this integral was not found by SymPy, Mathematica, RUBI, FriCAS, Maxima, or Giac in Desmond's tests.

A selection of indefinite integrals are summarized below for power functions

intz^rdz=(z^(r+1))/(r+1)+C
(23)
int(dz)/z=lnz+C
(24)
inta^zdz=(a^z)/(lna)+C,
(25)

trigonometric functions

intsinzdz=-cosz+C
(26)
intcoszdz=sinz+C
(27)
inttanzdz=ln(secz)+C
(28)
intcsczdz=ln(cscz-cotz)+C
(29)
=ln[tan(1/2z)]+C
(30)
=1/2ln((1-cosz)/(1+cosz))+C
(31)
intseczdz=ln(secz+tanz)+C
(32)
=ln[(cos(1/2z)+sin(1/2z))/(cos(1/2z)-sin(1/2z))]
(33)
=gd^(-1)(z)+C
(34)
intcotzdz=ln(sinz)+C,
(35)

combinations of trigonometric functions

intsin^2zdz=1/2z-1/4sin(2z)+C
(36)
intcos^2zdz=1/2z+1/4sin(2z)+C
(37)
inttan^2zdz=tanz-z+C
(38)
intsec^2zdz=tanz+C
(39)
intcsc^2zdz=-cotz+C
(40)
intcot^2zdz=-cotz-z+C
(41)
intsecztanzdz=secz+C,
(42)

inverse trigonometric functions

intcos^(-1)zdz=zcos^(-1)z-sqrt(1-z^2)+C
(43)
intsin^(-1)zdz=zsin^(-1)z+sqrt(1-z^2)+C
(44)
inttan^(-1)zdz=ztan^(-1)z-1/2ln(1+z^2)+C,
(45)

second-order rational functions and square roots

int(dz)/(sqrt(a^2-z^2))=sin^(-1)(z/a)+C
(46)
int(dz)/(sqrt(a^2-z^2))=-cos^(-1)(z/a)+C
(47)
int(dz)/(a^2+z^2)=1/atan^(-1)(z/a)+C
(48)
int(dz)/(a^2+z^2)=-1/acot^(-1)(z/a)+C
(49)
int(dz)/(zsqrt(z^2-a^2))=1/asec^(-1)(z/a)+C
(50)
int(dz)/(zsqrt(z^2-a^2))=-1/acsc^(-1)(z/a)+C,
(51)

Jacobi elliptic functions

intsn(u,k)du=k^(-1)ln[dn(u,k)-kcn(u,k)]+C
(52)
intcn(u,k)du=k^(-1)sin^(-1)[ksn(u,k)]+C
(53)
intdn(u,k)du=sin^(-1)sn(u,k)+C
(54)
=am(u,k)+C,
(55)

and the squares of Jacobi elliptic functions

intsn^2(u,k)du=(u-E(u,k))/(k^2)+C.
(56)

Here, sinz is the sine; cosz is the cosine; tanz is the tangent; cscz is the cosecant; secz is the secant; cotz is the cotangent; cos^(-1)z is the inverse cosine; sin^(-1)z is the inverse sine; tan^(-1) is the inverse tangent; snu, cnu, and dnu are Jacobi elliptic functions; am(u,k) is the Jacobi amplitude; E(u) is a complete elliptic integral of the second kind; and gd(z) is the Gudermannian. a is assumed to be real and positive, and k is the modulus.

To derive (28), let u=cosz, so du=-sinzdz and

inttanzdz=int(sinz)/(cosz)dz
(57)
=-int(du)/u
(58)
=-lnu+C
(59)
=-ln(cosz)+C
(60)
=ln(cosz)^(-1)+C
(61)
=ln(secz)+C.
(62)

To derive (29), let u=cscz-cotz, so du=(csc^2z-csczcotz)dz and

intcsczdz=intcscz(cscz-cotz)/(cscz-cotz)dz
(63)
=int(csc^2z-cotzcscz)/(cscz-cotz)dz
(64)
=int(du)/u
(65)
=lnu+C
(66)
=ln(cscz-cotz)+C.
(67)

To derive (32), let

u=secz+tanz,
(68)

so

du=(secztanz+sec^2z)dz
(69)

and

intseczdz=intsecz(secz+tanz)/(secz+tanz)dz
(70)
=int(sec^2z+secztanz)/(secz+tanz)dz
(71)
=int(du)/u=lnu+C
(72)
=ln(secz+tanz)+C.
(73)

To derive (35), let u=sinz, so du=coszdz and

intcotzdz=int(cosz)/(sinz)dz
(74)
=int(du)/u
(75)
=lnu+C
(76)
=ln(sinz)+C.
(77)

See also

Calculus, Constant of Integration, Contour Integral, Definite Integral, Fundamental Theorems of Calculus, Integral, Riemann Integral Explore this topic in the MathWorld classroom

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References

Boros, G. and Moll, V. Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge, England: Cambridge University Press, 2004.Desmond, H. "Exhaustive Symbolic Integration: Integration by Differentiation and the Landscape of Symbolic Integrability." 6 May 2026. https://arxiv.org/abs/2605.04978.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.Marchisotto, E. A. and Zakeri, G.-A. "An Invitation to Integration in Finite Terms." College Math. J. 25, 295-308, 1994.Ritt, J. F. Integration in Finite Terms: Liouville's Theory of Elementary Methods. New York: Columbia University Press, 1948.Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, 1993.

Referenced on Wolfram|Alpha

Indefinite Integral

Cite this as:

Weisstein, Eric W. "Indefinite Integral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/IndefiniteIntegral.html

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