Question

\[\int\left( x + 2 \right) \sqrt{x^2 + x + 1} \text{  dx }\]

Answer 1

\[\text{ Let I }= \int \left( x + 2 \right) \sqrt{x^2 + x + 1} \text{  dx }\]

\[\text{ Also,} x + 2 = \lambda\frac{d}{\text{  dx }} \left( x^2 + x + 1 \right) + \mu\]

\[ \Rightarrow x + 2 = \lambda\left( 2x + 1 \right) + \mu\]

\[ \Rightarrow x + 2 = \left( 2\lambda \right)x + \lambda + \mu\]

\[\text{Equating coefficient of like terms}\]

\[2\lambda = 1 \]

\[ \Rightarrow \lambda = \frac{1}{2}\]

\[\text{ And }\lambda + \mu = 2\]

\[ \Rightarrow \frac{1}{2} + \mu = 2\]

\[ \Rightarrow \mu = \frac{3}{2}\]

\[ \therefore I = \int \left[ \left( \frac{1}{2}\left( 2x + 1 \right) + \frac{3}{2} \right)\sqrt{x^2 + x + 1} \right]\text{  dx }\]

\[ = \frac{1}{2}\int \left( 2x + 1 \right) \sqrt{x^2 + x + 1} \text{  dx }+ \frac{3}{2}\int\sqrt{x^2 + x + 1} \text{  dx }\]

\[ = \frac{1}{2}\int\left( 2x + 1 \right) \sqrt{x^2 + x + 1}\text{  dx }+ \frac{3}{2}\int\sqrt{x^2 + x + \left( \frac{1}{2} \right)^2 - \left( \frac{1}{2} \right)^2 + 1}\text{  dx }\]

\[ = \frac{1}{2}\int \left( 2x + 1 \right) \sqrt{x^2 + x + 1} \text{  dx }+ \frac{3}{2} \int\sqrt{\left( x + \frac{1}{2} \right)^2 + \left( \frac{\sqrt{3}}{2} \right)^2} \text{  dx }\]

\[\text{ Let  x}^2 + x + 1 = t\]

\[ \Rightarrow \left( 2x + 1 \right)\text{  dx }= dt\]

\[\text{ Then, }\]

\[I = \frac{1}{2}\int \sqrt{t}\text{  dt } + \frac{3}{2} \int\sqrt{\left( x + \frac{1}{2} \right)^2 + \left( \frac{\sqrt{3}}{2} \right)^2} \text{  dx }\]

\[ = \frac{1}{2}\int t^\frac{1}{2} \text{  dt } + \frac{3}{2} \left[ \frac{x + \frac{1}{2}}{2} \sqrt{\left( x + \frac{1}{2} \right)^2 + \left( \frac{\sqrt{3}}{2} \right)^2} + \frac{3}{8}\text{ log }\left| \left( x + \frac{1}{2} \right) + \sqrt{\left( x + \frac{1}{2} \right)^2 + \left( \frac{\sqrt{3}}{2} \right)^2} \right| \right] + C\]

\[ = \frac{1}{2}\left[ \frac{t\frac{3}{2}}{\frac{3}{2}} \right] + \frac{3}{8}\left( 2x + 1 \right) \sqrt{x^2 + x + 1} + \frac{9}{16}\text{ log }\left| \left( x + \frac{1}{2} \right) + \sqrt{x^2 + x + 1} \right| + C\]

\[ = \frac{1}{3} \left( x^2 + x + 1 \right)^\frac{3}{2} + \frac{3}{8} \left( 2x + 1 \right) \sqrt{x^2 + x + 1} + \frac{9}{16}\text{ log }\left| \left( x + \frac{1}{2} \right) + \sqrt{x^2 + x + 1} \right| + C\]