Invert a function containing integration term

Consider this simple equation $$\tau(t)=\int\frac{dt}{a(t)},$$ where $$\tau$$ and $$a$$ are functions of $$t$$. Now, from this equation, how can I calculate $$a(\tau)$$ ?

• You should not use the same symbol for the limits of integration and the variable of integration. – Kavi Rama Murthy Mar 18 at 6:21
• Thanks a lot. I have deleted limits in my question. @Kavi Rama Murthy – Photon Mar 18 at 6:24

$$\tau'(t)=\frac 1 {a(t)}$$ so $$a(t)=\frac 1 {\tau'(t)}$$.
• Thanks a lot. But I need an equation of a as a function of $\tau$ i.e., the equation of $a(\tau$) not $a(t)$. @Kavi Rama Murthy – Photon Mar 18 at 6:27
• $\tau$ is a function of $t$, not a number. If you want to find the function $a\circ \tau$ the answer is $a(\tau (t))=1/\tau'(\tau (t))$. – Kavi Rama Murthy Mar 18 at 6:31