# An integral Equation

Let $$F(x)= \int_1^x \frac{1}{2 \sqrt{t}-1}dt$$ for all $x\geq1$. Then if $c>0$, there is a unique solution to the equation $F(x)=c$, $x>1$.

I calculated the integral but it didnt seem to help. What approach should I take?

• I'm worried by integrability at $t=1/4$, since $2\sqrt{\epsilon+1/4}-1 = 2\epsilon + O(\epsilon^2)$. Is it supposed to be a Cauchy PV integral ? – Stop hurting Monica May 2 '14 at 7:53
• Sorry it is supposed to be 1 I edited it – Paul Malinowski May 2 '14 at 8:00
• Ok, now do you want to prove there is a solution for all $c>0$, or to have a closed form for the solution? To prove there is a solution, just notice the integrand is positive (so the primitive is increasing), and by comparison with the integral of $1/\sqrt{x}$, the primitive is obviously not bounded when $x\rightarrow +\infty$. Then apply intermediate value theorem, since the primitive of a continuous function is continuous. – Stop hurting Monica May 2 '14 at 8:06

Note that $F'(x) = \frac{1}{2\sqrt{x} - 1}$. Since $x > 1$, what does that tell you about $F(x)$? What is $F(0)$? What is $\lim_{x \to \infty} F(x)$? How does the intermediate value theorem help?
• I don't understand. If $x>1$ then $0<F'(x)<1$, so $F$ is strictly increasing, and thus one-to-one? – Paul Malinowski May 2 '14 at 7:21
• Strictly increasing is one thing, but you also have to show that $F(x)$ is unbounded, since there are strictly increasing functions which are still bounded above and hence don't reach every positive real value. From there you can conclude that your function is one-to-one and has an inverse, if you feel that's obvious enough. – FlagCapper May 2 '14 at 7:32
• And $F(0) = -3/2$? – Paul Malinowski May 2 '14 at 7:34
Since the answers you received perfectly clarify your problem, le me focus on the solution. $$F(x)= \int_0^x \frac{1}{2 \sqrt{t}-1}dt=\frac{1}{2} \left(2 \sqrt{x}+\log \left(2 \sqrt{x}-1\right)-1\right)+\frac{1}{2} (1-i \pi )$$ and the unique solution of $F(x)=c$ is given by $$x=\frac{1}{4} \left(W\left(-e^{2 c-1}\right)+1\right)^2$$ where appears Lambert function.
• It would be interesting to see the derivation. At least to understand where this $i$ come from, since everything else is real... – Stop hurting Monica May 2 '14 at 7:45