Home » Blog » Evaluate the integral of (3-x2)1/2 Evaluate the integral of (3-x2)1/2 by RoRi December 17, 2015 Compute the following integral. To evaluate the integral we let Therefore we have Related
![\[ \int \sqrt{3-x^2} \, dx. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-e0414019830029eff936e84b0d8bbd6f_l3.png "Rendered by QuickLaTeX.com")
![\begin{align*} \int \sqrt{3-x^2} \, dx &= \int \sqrt{3-3 \sin^2 u} (\sqrt{3} \cos u \, du) \\[9pt] &= 3 \int \sqrt{1-\sin^2 u} (\cos u) \, du \\[9pt] &= 3 \int \cos^2 u \, du \\[9pt] &= \frac{3}{2} \int (\cos (2u) + 1) \, du \\[9pt] &= \frac{3}{4} \sin (2u) + \frac{3}{2} u + C \\[9pt] &= \frac{3}{4} \sin \left(2 \arcsin \frac{x}{\sqrt{3}} \right) + \frac{3}{2} \arcsin \frac{x}{\sqrt{3}} + C \\[9pt] &= \frac{3}{2} \sin \left( \arcsin \frac{x}{\sqrt{3}} \right) \cos \left( \arcsin \frac{x}{\sqrt{3}} \right) + \frac{3}{2} \arcsin \frac{x}{\sqrt{3}} + C \\[9pt] &= \frac{3}{2} \frac{x \sqrt{3-x^2}}{3} + \frac{3}{2} \arcsin \frac{x}{\sqrt{3}} + C \\[9pt] &= \frac{1}{2} x \sqrt{3-x^2} + \frac{3}{2} \arcsin \frac{x}{\sqrt{3}} + C. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-a268b4e05171532b60ea364b37298afa_l3.png "Rendered by QuickLaTeX.com")