<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Question parts:</span></div><ol class="i-list"><li><div class="list-item-paragraph">Show that \(\frac{3x^2}{(x+1)(3x^2+x+1)}\) can be expressed as </div><div class="list-item-paragraph" style="text-align: center">\(frac{A}{x+1} + \frac{Bx+C}{3x^2+x+1},\)</div><div class="list-item-paragraph">where \(A\), \(B\) and \(C\) are constants to be determined.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, find \(\displaystyle\int \frac{3x^2}{(x+1)(3x^2+x+1)} \, \mathrm{d}x\).</span><span class="inline-marks">[4]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">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.</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">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.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the exact area of \(R\).</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the volume of the solid generated when \(R\) is rotated through \(2\pi\) radians about the \(x\)-axis.</span><span class="inline-marks">[2]</span></div></li></ol></li></ol>

Show that \(\frac{3x^2}{(x+1)(3x^2+x+1)}\) can be expressed as\n\[\frac{A}{x+1} + \frac{Bx+C}{3x^2+x+1},\]\nwhere \(A\), \(B\) and \(C\) are constants to be determined.

Hence, find \(\displaystyle\displaystyle \displaystyle\int \frac{3x^2}{(x+1)(3x^2+x+1)} \, \mathrm{d}x\).

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.

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.

Find the volume of the solid generated when \(R\) is rotated through \(2\pi\) radians about the \(x\)-axis.

Techniques of  Integration

International Baccalaureate Higher Level Mathematics

2025 CJC P1 Q11

2025 CJC P1 Q11