Question

Solve this:

$$\begin{gathered} \begin{array}{c}\mathrm{Q}42·\int \frac{\mathrm{dx}}{x^2+\mathrm{x}+1}\mathrm{is}\mathrm{equal}\mathrm{to}\\ \left(\mathrm{A}\right)\frac{2}{\sqrt{3}}\sin -1\frac{2\mathrm{x}+1}{\sqrt{3}}+\mathrm{C}\\ \left(\mathrm{B}\right)\sin -\frac{12\mathrm{x}+1}{\sqrt{3}}+\mathrm{C}\\ \left(\mathrm{C}\right)\tan -1\frac{2\mathrm{x}+1}{\sqrt{3}}+\mathrm{C}\end{array} \\ \left(\mathrm{D}\right)\frac{2}{\sqrt{3}}{\tan }^{-1}\frac{2\mathrm{x}+1}{\sqrt{3}} \end{gathered}$$

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{I}=\int \frac{\mathrm{dx}}{{\mathrm{x}}^2+\mathrm{x}+1} \\ =\int \frac{\mathrm{dx}}{{\mathrm{x}}^2+\mathrm{x}+{\displaystyle \frac{1}{4}}+1-{\displaystyle \frac{1}{4}}} \\ =\int \frac{\mathrm{dx}}{{\left(\mathrm{x}+{\displaystyle \frac{1}{2}}\right)}^2+{\left({\displaystyle \frac{\sqrt{3}}{2}}\right)}^2} \\ =\frac{2}{\sqrt{3}}{\tan }^{-1}\left(\frac{2\mathrm{x}+1}{\sqrt{3}}\right)+\mathrm{C} \end{gathered}$$