Evaluate $\int_{0}^{l-a} \frac{dx}{\sqrt{(a+x)^2-a^2}} $

Question

Willing to evaluate $$\int_{0}^{l-a} \frac{dx}{\sqrt{(a+x)^2-a^2} }$$

The answer is given $ \cosh ^{-1} (l/a)$. Don't know how this can be achieved.

Please help

Answer 1

Hint: Substitute $$\sqrt{x^2+2ax}=x+t$$ then we get $$x=\frac{t^2}{2a-2t}$$ and $$dx=\frac{4at-2t^2}{(2a-2t)^2}dt$$ with the second line you Can express the term with the square root as a function in $t$ $$\sqrt{x^2+2ax}=t+\frac{t^2}{2a-2t}$$ $$\sqrt{x^2+2ax}=1/2\,{\frac {t \left( 2\,a-t \right) }{a-t}} $$