Single Variable CalculusIntegral FormulaIntegral with √[(x-a)(b-x)]On this pageIntegral with (x−a)(b−x)\sqrt{(x-a)(b-x)}(x−a)(b−x) 81 ∫1(x−a)(b−x) dx=2arcsinx−ab−a+C(a<b)\int{\frac{1}{\sqrt{(x-a)(b-x)}}}\ \mathrm{d}x=2\arcsin{\sqrt{\frac{x-a}{b-a}}}+C \quad (a<b)∫(x−a)(b−x)1 dx=2arcsinb−ax−a+C(a<b) Derivation 82 ∫(x−a)(b−x) dx=2x−a−b4(x−a)(b−x)+(b−a)24arcsinx−ab−a(a<b)\int{\sqrt{(x-a)(b-x)}}\ \mathrm{d}x=\frac{2x-a-b}{4} \sqrt{(x-a)(b-x)}+\frac{(b-a)^2}{4}\arcsin{\sqrt{\frac{x-a}{b-a}}} \quad (a<b)∫(x−a)(b−x) dx=42x−a−b(x−a)(b−x)+4(b−a)2arcsinb−ax−a(a<b) Derivation