List of Functions Without Antiderivatives
In calculus, every single-variable function composed of elementary functions has a derivative, but not every such function can be integrated exactly over any arbitrary integral, i.e., you cannot express the antiderivative in terms of elementary functions to find the exact integral for any interval. Exact integrals forms are also called closed-form expressions.
Elementary functions include polynomials, rational functions, radical expressions, exponential functions, logarithms, trig functions, and inverse trig functions. For example, consider the complicated function
f(x) = [Ln(x)*e^sqrt(x)]/[Ln(1 +e^x) + x^3]
The derivative of f(x) can be found by applying the product, quotient, and chain rules for differentiation. The expression for f'(x) is shown in the image below. As you can see it is rather complicated, but nevertheless it can be expressed in terms of elementary functions.
However, the antiderivative of f(x) cannot be expressed in this way. This is an example of a function that cannot be integrated in the usual way; it has no closed form expression in terms of elementary functions. To find the integral of this function over the interval x = 9 to x = 4π you have to use numerical methods such as Reimann sums or approximate antiderivatives. Graphing calculators use Reimann sums. Taylor series and asymptotic series for functions can be manipulated to produce approximate antiderivatives.
There are infinitely many functions without antiderivatives in terms of elementary functions. If you are able to recognize them by sight, it can save you a lot of grief when solving challenging integral calculus problems. The following is not an exhaustive list, but covers many of the simplest examples of non-integrable functions that you may encounter in Calc II.
Here is a list of basic exponential forms that have no antiderivatives
- e^(x^2) and e^(-x^2)
- e^(x^3) and e^(-x^3)
- More generally, e^(x^n) and e^(-x^n) are non-integrable for all integers n greater than 1.
- e^(1/x) and e^(-1/x) (See link for approximate integration technique.)
- e^(1/x^2) and e^(-1/x^2)
- More generally, e^(1/x^n) and e^(-1/x^n) are non integrable for all positive integers n.
Here are other more complicated examples exponential functions that cannot be integrated. In general, the more complicated f(x) is, the less likely e^f(x) is integrable.
- e^(e^x), e^(-e^x), and e^(e^-x) (See link for approximate integral with series.)
- More generally, any function of the form (e^x)/p(x), where p(x) is a polynomial, cannot be integrated.
- x/(1 + e^x)
- x/(x + e^x)
- 1/(x + e^x)
- More generally, any function of the form [p(x) + e^x]/[q(x) + e^x], where p(x) and q(x) are polynomials, cannot be integrated, except in the rare case when p(x) is the derivative of q(x). In that case, use the substitution u = q(x) + e^x.
- e^sin(x), e^cos(x), and e^tan(x)
- e^arcsin(Ln(x)), e^arccos(Ln(x))
The square roots, cube roots, fourth roots, etc. of certain functions often have antiderivatives that cannot be expressed in terms of elementary functions. The most notable belong to the family of functions known as Lamé curves:
L(x) = (1 ± x^m)^(1/n)
where m and n are numbers greater than 1. The notable exception is m = n = 2, which produces a circle, a curve that is integrable. Non-integrable examples in this family include
- (1 - x^3)^(1/3)
- (1 - x^4)^(1/2)
- (1 + x^2)^(1/4)
Other radical expressions that are not integrable are
- sqrt(1 ± Ln(x)) = (1 ± Ln(x))^(1/2)
- sqrt(x ± Ln(x))
- sqrt(x ± e^x)
- sqrt(p(x)), where p(x) is a polynomial of degree greater than 2.
- More generally, p(x)^(1/n) is not integrable for any integer n greater than 2 and any polynomial with a degree greater than 2, except in some cases where p(x) is a perfect power or the radical can be reduced to a simpler form.
- any function of the form 1/(1 ± x^m)^(1/n) = (1 ± x^m)^(-1/n) where m and n are greater than 1, except the case where m = n = 2.
- sqrt(a ± sin(x)), except when a = 1
- sqrt(a ± cos(x)), except when a = 1
- sqrt(x)e^x = (x^(1/2))e^x
- More generally, the function (x^c)e^x is not integrable unless c is a non-negative integer.
- More generally, any function of the form (x^b)sin(x) or (x^b)cos(x) is non-integrable unless b is a non-negative integer.
Although functions of the form q(x)*Ln(p(x)) -- where p(x) and q(x) are polynomials -- are integrable, most other compositions of log functions are not. Some of the most basic non-integrable forms are
- Ln(1 ± e^x) (See link for approximate integral with series.)
- Ln(p(x) ± e^x), where p(x) is any polynomial
- 1/[Ln(x) + x]
- More generally if p(x) is a polynomial, then p(x)/Ln(x) is non-integrable.
- Ln(x)/(x ± 1)
- More generally, any function of the form Ln(x)/p(x) is non-integrable unless p(x) is a pure power function x^n.
- Functions of the form Ln(p(x))*Ln(q(x)) are non-integrable, unless p(x) and q(x) are pure powers of the same linear function. For example, p(x) = (2x-5)^3 and q(x) = (2x-5)^7.
- Functions of the form Ln(p(x))/Ln(q(x)) are non-integrable except in the trivial case when p(x) and q(x) are pure powers of the same polynomial. For example, p(x) = (x^2 + 3)^2 and q(x) = (x^2 + 3)^9.
- If a function with logarithm base-e (natural logarithm) "Ln" is not integrable in the usual way, then replacing it with a logarithm of a different base won't change that fact.
Curious exceptions: Although the functions g(x) = Ln(x)*e^x and h(x) = x*Ln(x)*e^x are both non-integrable in the usual way, their sum
g(x) + h(x) = (x+1)*Ln(x)*e^x
does in fact have an antiderivative:
∫ (x+1)*Ln(x)*e^x dx = [x*Ln(x) - 1]e^x + c
This illustrates that the sum (or difference, product, or quotient) of two functions without elementary antiderivatives may yet have a surprisingly simple antiderivative.
Trigonometric functions have many of the same properties of exponential functions because they can be defined in terms of e^x extended to complex numbers. The identities are
sin(x) = [e^(ix) - e^(-ix)] / [2i]
cos(x) = [e^(ix) + e^(-ix)] / 2
tan(x) = -i*[e^(ix) - e^(-ix)] / [e^(ix) + e^(-ix)]
Forms of exponential functions that are non-integrable are similar to forms of trig functions that are non-integrable. Some basic examples include
- sin(x^2), cos(x^2), and tan(x^2)
- sin(1/x), cos(1/x), and tan(1/x)
- More generally, sin(x^d), cos(x^d) and tan(x^d) are non-integrable unless d is the reciprocal of a positive integer (or trivially, d = 0).
- sin(x)/x, cos(x)/x, and tan(x)/x (See link for approximate integral.)
- 1/(sin(x) + x), 1/(cos(x) + x), and 1/(tan(x) + x)
- More generally, the function (x^p)*tan(x) is non-integrable except in the trivial case where p = 0.
- sqrt(sin(x)) and sqrt(cos(x)) (See link for approximate integral.)
- sin(sin(x)), cos(cos(x)), tan(tan(x))
- sin(cos(x)), cos(sin(x), tan(sin(x)), tan(cos(x)), sin(tan(x)), and cos(tan(x))
- sin(e^x), cos(e^x), and tan(e^x)
- x/sin(x), x/cos(x), and x/tan(x)
Just as trig functions are related to exponential functions, so are inverse trig functions related to logarithmic functions. This means many non-integrable inverse trig functions are similar to logarithmic functions that have no antiderivatives. Some examples of non-integrable inverse trig functions include
- 1/arcsin(x), 1/arccos(x), and 1/arctan(x)
- arcsin(x^2) and arccos(x^2)
- arcsin(x)/x, arccos(x)/x, and arctan(x)/x
- x/arcsin(x), x/arccos(x), and x/arctan(x)
- arcsin(e^x), arccos(e^x), and arctan(e^x)
- arcsin(Ln(x)), arccos(Ln(x)), and arctan(Ln(x))
Notable Exceptions with Definite Integrals
Although some functions cannot be integrated with elementary antiderivatives, many of them can be evaluated in terms of well-known mathematical constants for certain definite integrals. Perhaps the most famous example is the integral
∫ e^(-x^2) dx
evaluated from x = -infinity to x = infinity, which is equal to exactly sqrt(π). This was proven by Gauss using a clever change of variables for the three-dimensional version of the function. Here are some more definite integrals that have surprising definite integrals.
∫ Ln(x)*Ln(1-x) dx, [0, 1] = (12 - π^2)/6 (shown in graph below)
∫ Ln(x)*e^(-x) dx, [0, ∞) = -γ, where γ is the Euler-Mascheroni constant.
∫ sqrt(x)*e^(-x) dx, [0, ∞) = sqrt(π)/2
∫ Ln(1 + e^(-x)) dx, [0, ∞) = (π^2)/12