Theres a hint to use $x=5+5\sin{t}$. Ok, but how do I know what substitution to use if a hint wasn't given? Is it "trivial" or perhaps, its very unlikely that that will appear?
Anyways, I did:
$\int \sqrt{10(5+5\sin{t}) - (5+2\sin{t})^2} dx \\ = \int \sqrt{50+50\sin{t} - (25+50\sin{t} + 25\sin^2{t})} dx\\ = \int \sqrt{ 25-25\sin^2{t} } dx \\ = 5 \int \sqrt{1-\sin^2{t}} dx \\ = 5\sin^{-1}{\sin{t}} \\ = 5t \\ = 5 \sin^{-1}{\frac{x-5}{5}}$
But the answer was:
$\frac{25}{2}\sin^{-1}{\frac{x-5}{5}}+\frac{x-5}{2}\sqrt{10x-x^2}+c$
What did I do wrong? Or is the answer wrong perhaps?