Question

\[\int\frac{x + 1}{x^2 + x + 3} dx\]

Answer 1

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

Comparing Coefficients of like powers of x

\[2A = 1\]
\[A = \frac{1}{2}\]
\[A + B = 1\]
\[\frac{1}{2} + B = 1\]
\[B = \frac{1}{2}\]
\[\left( x + 1 \right) = \frac{1}{2} \left( 2x + 1 \right) + \frac{1}{2}\]

\[Now, \int\frac{\left( x + 1 \right) dx}{x^2 + x + 3}\]
\[ = \int\frac{\frac{1}{2} \left( 2x + 1 \right)dx}{x^2 + x + 3} + \frac{1}{2}\int\frac{dx}{x^2 + x + 3}\]
\[ = \frac{1}{2}\int\frac{\left( 2x + 1 \right)dx}{x^2 + x + 3} + \frac{1}{2}\int\frac{dx}{x^2 + x + \left( \frac{1}{2} \right)^2 - \left( \frac{1}{2} \right)^2 + 3}\]
\[ = \frac{1}{2}\int\frac{\left( 2x + 1 \right)dx}{x^2 + x + 3} + \frac{1}{2}\int\frac{dx}{\left( x + \frac{1}{2} \right)^2 + 3 - \frac{1}{4}}\]
\[ = \frac{1}{2}\int\frac{\left( 2x + 1 \right) dx}{x^2 + x + 3} + \frac{1}{2}\int\frac{dx}{\left( x + \frac{1}{2} \right)^2 + \left( \frac{\sqrt{11}}{2} \right)^2}\]
\[ = \frac{1}{2} \text{ log }\left| x^2 + x + 3 \right| + \frac{1}{2} \times \frac{2}{\sqrt{11}} \text{ tan}^{- 1} \left( \frac{x + \frac{1}{2}}{\frac{\sqrt{11}}{2}} \right) + C\]
\[ = \frac{1}{2} \text{ log }\left| x^2 + x + 3 \right| + \frac{1}{\sqrt{11}} \text{ tan}^{- 1} \left( \frac{2x + 1}{\sqrt{11}} \right) + C\]