What is the name of an equation, where the unknown is one of the limits of integration? Is there a theory that studies such equations, standard methods of solution?

The simplest example is the equation defining a median in statistics $\int_{-\infty}^x f(y)dy = 0.5 $, where $x$ is the unknown.

  • $\begingroup$ You have tagged one. Equations of involving integrals like this one are called integral equations, and they are a large field of study related to differential equations. $\endgroup$
    – Plutoro
    Jun 26, 2015 at 22:12
  • 1
    $\begingroup$ @AlexS Not really. In an integral equation the unknown is a function appearing in the integrand. Here it seems the integrand is known, the unknown is an endpoint of the interval of integration. $\endgroup$ Jun 26, 2015 at 22:26
  • $\begingroup$ @RobertIsrael good call. Thanks for clarifying. $\endgroup$
    – Plutoro
    Jun 26, 2015 at 22:27

2 Answers 2


me again. I was very tired when I asked the question, the answer is trivial

$ \int_a^b f(x)dx = \Phi(b) - \Phi(a) $, where $\Phi(x) = \int f(x) dx $ is the corresponding definite integral.

So the problem boils down to being able to compute a definite integral, and then being able so solve the resulting ordinary equation.

  • $\begingroup$ I think you mean indefinite integral, not definite integral. $\endgroup$
    – asherbret
    Sep 26, 2018 at 17:22

You need to implement certain numerical methods in order to determine the value of limit approximately. Define, \begin{equation} h(x) = 0.5 - \int_{-\infty}^{x} f(z) dz \end{equation} Now all you need to do is apply Newton-Raphson or any other standard root finding algorithm to figure out the value of $x$ which can be done in the following way.

  1. Start with some initial guess $x_{0}$
  2. Proceed iteratively using $x_{n+1} = x_{n} + \frac{0.5 - \int_{-\infty}^{x_{n}} f(x) dx}{f(x_{n})}$
  3. Stop the iterations when $|x_{n+1} - x_{n}| < \epsilon $, where $\epsilon$ is specified by the user.

The above process would provide a reasonably close approximation to the required value of $x$.

  • $\begingroup$ Cool idea, but it's not enough. Newton-Raphson does not converge for functions with local extrema. But your idea have me a new one using binary section. Map the infinite domain to finite using e.g. tangent transform. Split finite interval into eg 100 smaller intervals and compute integral fit each piece. For each piece sum up all the pieces before it. As soon as the sign of the integral flips, you know that the solution is inside that piece, repeat recursively. Perhaps Metropolis Hastings can do even better, not sure $\endgroup$
    – Alyosha
    Dec 13, 2016 at 16:50
  • $\begingroup$ Ah sorry, I mixed up with optimisation. Newton-Raphson would work too, just need educated guesses for x0 to find all roots $\endgroup$
    – Alyosha
    Dec 13, 2016 at 16:53

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