Show that $\frac{3x^2}{(x+1)(3x^2+x+1)}$ can be expressed as \[\frac{A}{x+1} + \frac{Bx+C}{3x^2+x+

Question

Show that $\frac{3x^2}{(x+1)(3x^2+x+1)}$ can be expressed as $$\frac{A}{x+1} + \frac{Bx+C}{3x^2+x+1}$$ where $A$, $B$ and $C$ are constants to be determined.[2] Hence, find $\displaystyle\int \frac{3x^2}{(x+1)(3x^2+x+1)} \, \mathrm{d}x$.[4] The region $R$ is bounded by the curve $y = \frac{3x^2}{(x+1)(3x^2+x+1)}$, the line $x=4$ and the $x$-axis. Questions on region $R$: Sketch the graph of $y = \frac{3x^2}{(x+1)(3x^2+x+1)}$ for $x \ge 0$, giving the coordinates of any intercepts with the axes and the equations of any asymptotes. Shade the region $R$ on your sketch.[3] Find the exact area of $R$.[3] Find the volume of the solid generated when $R$ is rotated through $2\pi$ radians about the $x$-axis.[2]

Accepted Answer

(a)(i)A =1, B = -1 (a)(ii)$\ln|x+1|-\frac{2}{\sqrt11}\tan^{-1}(\frac{6x+1}{\sqrt11})+C$ (b)(ii)Area of R =$\ln 5 +\frac{2}{\sqrt11}[\tan^{-1}(\frac{1}{\sqrt11})-tan^{-1}(\frac{25}{\sqrt11})]$ (b)(iii)0.715 (3 sf)

2025 CJC P1 Q11

2025 CJC P1 Q11