Since _Mathematica_ is a really powerful symbolic computing system I recommend a to proceed symbolically as far as we can.

We define a function `curve` depending on `t` and four real positive parameters `a`, `b`, `c`, `d`: 

    curve[t_, a_, b_, c_, d_] := a Exp[-((b (c - d t)^2)/t)]/Sqrt[t]

Now _Mathematica_ can evaluate such an integral symbolically:

    integral[x0_, x_, a_, b_, c_, d_] =
      Integrate[ curve[ t, a, b, c, d], { t, x0, x}, 
                 Assumptions -> Join[{ x0 <= x, x >= 0}, 
                                      Thread[# > 0 &[{a, b, c, d}]]]]
>     -((a Sqrt[ Pi] (Erf[Sqrt[b/x] (c - d x)] - E^(4 b c d) Erf[Sqrt[b/x] (c + d x)] 
    + Erf[(b (-c + d x0))/Sqrt[b x0]] 
    + E^(4 b c d) Erf[(b (c + d x0))/Sqrt[b x0]]))/(2 Sqrt[b] d))

We have assumed lower limit of integration `x0`, if we had assumed `x0 == 0` the system couldn't provide a symbolic result yielding only an unsatisfactory warning:

>     Integrate::idiv: "Integral of E^(-((b (c-d t)^2)/t))/Sqrt[t] does not converge on {0, x}."

However our integral can be evaluated in the limit as `x0 -> 0`:

    Limit[ integral[x0, x, a, b, c, d], x0 -> 0, 
           Assumptions -> {x >= 0, a > 0, b > 0, c > 0, d > 0}]
>      (a Sqrt[ Pi] (1 - E^(4 b c d) - Erf[ Sqrt[b/x] (c - d x)]
    + E^(4 b c d) Erf[ Sqrt[b/x] (c + d x)]))/(2 Sqrt[b] d)

Now you can solve your equation with `Solve` providing rational parameters (possibly using appropriate options), otherwise for real parameters use `NSolve` or `FindRoot`. Now, it is reasonable to search for numeric solutions since there are `Erf` functions involved. Your original parameters haven't been justified and it might be necessary it play with `AccuracyGoal` and `PrecissionGoal`. E.g. we can easily find the upper limit of integration `x` these reasonable parameters:

    FindRoot[ area[x, 0.2, 0.3, 6.1, 3.1] == 0.01, {x, 0.5}, 
              AccuracyGoal -> 5, PrecisionGoal -> 5]
>     {x -> 1.26325}