Integrate $\int \frac {dx}{\sqrt {(x-a)(x-b)}}$ where $b>a$ Integrate: $\displaystyle\int \dfrac {dx}{\sqrt { (x-a)(x-b)}}$ where $b>a$
My Attempt:
$$\int \dfrac {dx}{\sqrt {(x-a)(x-b)}}$$
Put $x-a=t^2$
$$dx=2t\,dt$$
Now,
\begin{align}
&=\int \dfrac {2t\,dt}{\sqrt {t^2(a+t^2-b)}}\\
&=\int \dfrac {2\,dt}{\sqrt {a-b+t^2}}
\end{align}

Proceeding further from where you left 
$$
\int \dfrac {2\,}{\sqrt {a-b+t^2}}\ dt
$$
$$
\int \dfrac {2}{\sqrt {(t)^2 + (\sqrt{a-b}) ^2}}\ dt
$$
Using
$$
\int \frac{1}{\sqrt{x^2 + a ^ 2}} = \log({x+\sqrt{x^2+a^2}})
$$
It becomes 
$$
2 \cdot {\log (t+\sqrt{t^2 + (\sqrt{a-b}) ^2})}
$$
Then substitute back the value of t .
And you get the required result .
