Chapter 8: Techniques of Integration - Section 8.1 - Using Basic Integration Formulas - Exercises 8.1 - Page 448: 5

$$\int \frac{1-x}{\sqrt{1-x^{2}}}dx=arcsin\,x+\sqrt{1-x^{2}}+C$$

$$\int \frac{1-x}{\sqrt{1-x^{2}}}dx=\int \frac{1}{\sqrt{1-x^{2}}}dx-\int \frac{x}{\sqrt{1-x^{2}}}dx$$ $$=arcsin\,x+C_{1}+\frac{1}{2}\int \frac{d(1-x^{2})}{\sqrt{1-x^{2}}}$$ $$=arcsin\,x+C_{1}+\frac{1}{2}\int (1-x^{2})^{-\frac{1}{2}} d(1-x^{2})$$ $$=arcsin\,x+\sqrt{1-x^{2}}+C$$