[Math] Integration by parts $ \int \sqrt{x-x^2} dx $

Question

I have to integrate $$ \int \sqrt{x-x^2} dx $$
The answer on my textbook is $ \frac 14 \left( \arcsin(\sqrt x)-(1-2x)\sqrt{x-x^2} \right) $ but I want to solve this by myself so can you please give me a clue ? 🙂 Thank you.

Answer 1

Hint: Complete the square: $$\int\sqrt{x-x^2}dx=\int\sqrt{-x^2+2\frac x2-\frac14+\frac14}dx=\int\sqrt{-\left(x-\frac 12\right)^2+\frac14}dx$$ Set $u=x-\frac 12$ to obtain $$\int\sqrt{-u^2+\frac14}du$$ I think you now how to deal with this one.