Some Important Integrals
(i) \(\sqrt{{{a}^{2}}-{{x}^{2}}}dx=\frac{1}{2}x\sqrt{{{a}^{2}}-{{x}^{2}}}+\frac{1}{2}{{a}^{2}}{{\sin }^{-1}}\left( \frac{x}{a} \right)+C\).
(ii) \(\int \sqrt{{{a}^{2}}+{{x}^{2}}}dx=\frac{1}{2}x\sqrt{{{a}^{2}}+{{x}^{2}}}+\frac{1}{2}{{a}^{2}}\log \left| x+\sqrt{{{a}^{2}}+{{x}^{2}}} \right|+C\).
(iii) \(\int \sqrt{{{x}^{2}}-{{a}^{2}}}dx=\frac{1}{2}x\sqrt{{{x}^{2}}-{{a}^{2}}}-\frac{1}{2}{{a}^{2}}\log \left| x+\sqrt{{{x}^{2}}-{{a}^{2}}} \right|+C\).
Example: \(\int{\sqrt{{{x}^{2}}+4x+6}}.dx\) is equal to ?
Solution: Given,
\(\int{\sqrt{{{x}^{2}}+4x+6}}.dx\),
Here, we can convert the integrand into some standard integrand of the form \(\sqrt{{{x}^{2}}\pm {{a}^{2}}}\) and then integrate.
Let, \(\int{\sqrt{{{x}^{2}}+4x+6}}.dx\),
\(\int{\sqrt{{{x}^{2}}+4x+{{2}^{2}}+6-4}}.dx\),
\(\int{\sqrt{{{(x+2)}^{2}}+{{(\sqrt{2})}^{2}}}}.dx\) (∵ \(\int \sqrt{{{a}^{2}}+{{x}^{2}}}dx=\frac{1}{2}x\sqrt{{{a}^{2}}+{{x}^{2}}}+\frac{1}{2}{{a}^{2}}\log \left| x+\sqrt{{{a}^{2}}+{{x}^{2}}} \right|+C\)),
\(=\frac{x+2}{2}\sqrt{{{x}^{2}}+4x+6}+\frac{2}{2}\log |(x+2)+\sqrt{{{x}^{2}}+4x+6}|+C\),
\(=\frac{x+2}{2}\sqrt{{{x}^{2}}+4x+6}+\log |(x+2)+\sqrt{{{x}^{2}}+4x+6}|+C\).