Evaluate the following integral
![\[ \int \sqrt{1-x^2} \, dx. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-65a53c2f9371fcef900e74f6bfb4d521_l3.png "Rendered by QuickLaTeX.com")
We integrate by parts, letting
Then we have
![\begin{align*} &&\int \sqrt{1-x^2} \, dx &= x \sqrt{1-x^2} + \int \frac{x^2}{\sqrt{1-x^2}} \, dx \\[9pt] \implies && \int \sqrt{1-x^2} \, dx &= x \sqrt{1-x^2} + \int \frac{1}{\sqrt{1-x^2}} \, dx - \int \frac{1-x^2}{\sqrt{1-x^2}} \, dx \\[9pt] \implies && 2 \int \sqrt{1-x^2} \, dx &= x \sqrt{1-x^2} + \arcsin x + C \\[9pt] \implies && \int \sqrt{1-x^2} \, dx & = \frac{1}{2} \left( \arcsin x + x \sqrt{1-x^2} \right) + C. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-07eee6c80b3b1bc53848e341238cbba9_l3.png "Rendered by QuickLaTeX.com")
(As is often the case, there are other ways to do this. One might want to try a substitution, letting 
.)
Evaluate the following integral
We integrate by parts, letting
Then we have
(As is often the case, there are other ways to do this. One might want to try a substitution, letting .)