# How to invert an integral equation

There have been numerous times when I've needed to invert an integral equation, i.e. I have something like $$f(x) = g_1(x)\int_{0}^x g_2(x') dx'$$ for arbitrary functions $g_1$ and $g_2$, and I would like to find $x$ for a given $f$. The way I've gotten around this is just making some sort of table of f(x) and x with spacing up to some required precision. Is there a more efficient method of doing it (numerically or analytically)?

When I try to use NSolve[ f(x) == a, x ] mathematica complains NIntegrate::nlim: x = x is not a valid limit of integration.

My interpretation of the question is that you want to find $x$ for given $f$, $g_2$ and $g_1$. Then just define $F=f/g_1$ and differentiate with respect to $x$ on both sides:

$\frac{d}{dx}F(x)=g_2(x)$

Now solve this equation for $x$. There's no integration involved.

• Just implemented this and it works well. Using a simple finite-difference routine to calculate the derivative of 'F' is computationally trivial compared to the iterative numerical integrations. – DilithiumMatrix Apr 22 '12 at 4:59
• I tried to implement this using ND and FindRoot, but I am getting strange problems. Maybe I need to ask a separate question. – Santi Aug 15 '14 at 11:59
• Here is the separate question: mathematica.stackexchange.com/questions/57435/… – Santi Aug 19 '14 at 9:53

f[x_?NumericQ] := f[x] = g1[x] NIntegrate[g2[t], {t, 0, x}];