Question

Solve $\int \frac{dt}{t\sqrt{t^2-1}}$

Answer 1

Consider $\int \frac{dt}{t\sqrt{t^2-1}}$

Let $u=\sqrt{t^2-1}$

$\Rightarrow du=\frac{1}{2\sqrt{t^2-1}}\times 2tdt=\frac{t}{\sqrt{t^2-1}}dt$

$\Rightarrow \frac{dt}{\sqrt{t^2-1}}=\frac{du}{t}$

$\Rightarrow \frac{dt}{t\sqrt{t^2-1}}=\frac{du}{t^2}$

$\therefore \int \frac{dt}{t\sqrt{t^2-1}}=\int \frac{du}{t^2}$

We have $u=\sqrt{t^2-1}\Rightarrow t^2=u^2+1$

$\therefore \int \frac{dt}{t\sqrt{t^2-1}}=\int \frac{du}{1+u^2}=ta{n}^{-1}u$

Substitute back $u=\sqrt{t^2-1}$

$\int \frac{dt}{t\sqrt{t^2-1}}=ta{n}^{-1}\left(\sqrt{t^2-1}\right)+c$