# Some Important Integrals

## 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,  $$\int{\sqrt{{{x}^{2}}+4x+6}}.dx$$,

$$\int{\sqrt{{{x}^{2}}+4x+{{2}^{2}}+6-4}}.dx$$,

$$\int{\sqrt{{{(x+2)}^{2}}+{{(\sqrt{2})}^{2}}}}.dx$$ (∵ $$\int \sqrt{{{a}^{2}}+{{x}^{2}}}dx=\frac{1}{2}x\sqrt{{{a}^{2}}+{{x}^{2}}}+\frac{1}{2}{{a}^{2}}\log \left| x+\sqrt{{{a}^{2}}+{{x}^{2}}} \right|+C$$),

$$=\frac{x+2}{2}\sqrt{{{x}^{2}}+4x+6}+\frac{2}{2}\log |(x+2)+\sqrt{{{x}^{2}}+4x+6}|+C$$,

$$=\frac{x+2}{2}\sqrt{{{x}^{2}}+4x+6}+\log |(x+2)+\sqrt{{{x}^{2}}+4x+6}|+C$$.