Question

Find : \[\int\left( 2x + 5 \right)\sqrt{10 - 4x - 3 x^2}dx\] .

Answer 1

\[2x + 5 = \lambda\frac{d}{dx}\left( 10 - 4x - 3 x^2 \right) + \mu\]

\[ \Rightarrow 2x + 5 = \lambda\left( - 4 - 6x \right) + \mu\]

\[ \Rightarrow 2x + 5 = - 6\lambda x - 4\lambda + \mu\]

\[ \Rightarrow - 6\lambda = 2, - 4\lambda + \mu = 5\]

\[ \Rightarrow \lambda = - \frac{1}{3}, \mu = \frac{11}{3}\]

\[\therefore I = \int\left( 2x + 5 \right)\sqrt{10 - 4x - 3 x^2}dx\]

\[\Rightarrow I = \int\left[ - \frac{1}{3}\left( - 4 - 6x \right) + \frac{11}{3} \right]\sqrt{10 - 4x - 3 x^2}dx\]

\[ \Rightarrow I = - \frac{1}{3}\int\left( - 4 - 6x \right)\sqrt{10 - 4x - 3 x^2}dx + \frac{11}{3}\int\sqrt{10 - 4x - 3 x^2}dx \]

\[Put 10 - 4x - 3 x^2 = t in the first integral . \]

\[ \therefore \left( - 4 - 6x \right)dx = dt\]

\[I = - \frac{1}{3}\int\sqrt{t}dt + \frac{11}{3}\int\sqrt{- 3\left( x^2 + \frac{4}{3}x - \frac{10}{3} \right)}dx\]

\[ \Rightarrow I = - \frac{1}{3} \times \frac{2}{3} t^\frac{3}{2} + C_1 + \frac{11}{3}\int\sqrt{- 3\left( x^2 + \frac{4}{3}x - \frac{10}{3} + \frac{4}{9} - \frac{4}{9} \right)}dx\]

\[ \Rightarrow I = - \frac{2}{9} \left( 10 - 4x - 3 x^2 \right)^\frac{3}{2} + C_1 + \frac{11}{3}\int\sqrt{- 3\left[ \left( x + \frac{2}{3} \right)^2 - \left( \frac{\sqrt{34}}{3} \right)^2 \right]}dx\]

\[ \Rightarrow I = - \frac{2}{9} \left( 10 - 4x - 3 x^2 \right)^\frac{3}{2} + C_1 + \frac{11 \times \sqrt{3}}{3}\int\sqrt{\left( \frac{\sqrt{34}}{3} \right)^2 - \left( x + \frac{2}{3} \right)^2}dx\]

\[ \Rightarrow I = - \frac{2}{9} \left( 10 - 4x - 3 x^2 \right)^\frac{3}{2} + C_1 \]

\[ + \frac{11\sqrt{3}}{3}\left[ \frac{1}{2}\left( x + \frac{2}{3} \right)\sqrt{\frac{34}{9} - \left( x + \frac{2}{3} \right)^2} + \frac{\frac{34}{9}}{2} \sin^{- 1} \left( \frac{x + \frac{2}{3}}{\frac{\sqrt{34}}{3}} \right) + C_2 \right]\]

\[ \Rightarrow I = - \frac{2}{9} \left( 10 - 4x - 3 x^2 \right)^\frac{3}{2} + \frac{11}{2\sqrt{3}}\left( x + \frac{2}{3} \right)\sqrt{\left( \frac{34}{9} \right) - \left( x + \frac{2}{3} \right)^2} + \frac{187}{9\sqrt{3}} \sin^{- 1} \left( \frac{3x + 2}{\sqrt{34}} \right) + C\]