Question:

<p>Write $(2x - 4) = \ell (\text{d.c. of } 4 + 3x - x^2) + m$ or $(2x - 4) = \ell(-2x + 3) + m$.</p><p>Comparing the coefficient, we get</p><p>$-2\ell = 2, 3\ell + m = -4 \Rightarrow \ell = -1, m = -1$</p><p>Hence $I = \int (-2x+3)\sqrt{4+3x-x^2} dx - \int \sqrt{4+3x-x^2} dx$</p><p>$= -\frac{2}{3}(4+3x-x^2)^{3/2} - \int \sqrt{\left(\frac{5}{2}\right)^2 - \left(x-\frac{3}{2}\right)^2} dx$</p><p>$= -\frac{2}{3}(4+3x-x^2)^{3/2} - \left[ \frac{x-\frac{3}{2}}{2} \sqrt{\left(\frac{5}{2}\right)^2 - \left(x-\frac{3}{2}\right)^2} + \frac{25}{8} \sin^{-1} \left( \frac{2x-3}{3} \right) \right] + c$</p>