June 2025 Paper 2 Q2
2. Find\[\int\left(x^4 - 6x^{\frac{1}{2}} - 3\right)\mathrm{d}x\]giving the answer in simplest form. (4)
| Scheme | Marks | AO |
|---|---|---|
| \(x^4 \rightarrow \ldots x^5\) or \(x^{\frac{1}{2}} \rightarrow \ldots x^{\frac{3}{2}}\) or \(-3 \rightarrow \ldots x\) | M1 | 1.1a |
| Any of \(\dfrac{1}{5}x^5\) or \(-\dfrac{6}{\left(\frac{3}{2}\right)}x^{\frac{3}{2}}\) or \(-3x\) | A1 | 1.1b |
| Any two of \(\dfrac{1}{5}x^5\) or \(-\dfrac{6}{\left(\frac{3}{2}\right)}x^{\frac{3}{2}}\) or \(-3x\) | A1 | 1.1b |
| \(\dfrac{1}{5}x^5 - 4x^{\frac{3}{2}} - 3x + c\) | A1 | 1.1b |
| (4) | ||
| (4 marks) |
Notes
M1: For increasing any power by one. Score for \(x^n \rightarrow x^{n+1}\) in any term, including, \(-3 \rightarrow \ldots x\) where … is a constant, but not for + \(c\). Allow the indices to be unprocessed, e.g., \(x^{4+1}\)
A1: One correct term which may be unsimplified and indices may be unprocessed.
Condone e.g. \(-3x^1\) or \(-\dfrac{6}{\left(\frac{1}{2} + 1\right)}x^{\frac{1}{2} + 1}\) for this mark. Not scored for + \(c\)
A1: Two correct terms which may be unsimplified but indices must be processed.
Condone \(-3x^1\) for this mark. Not scored for + \(c\)
A1: cao Requires all terms simplified and + \(c\)
Ignore the LHS i.e. ignore what they call their integral.
Allow \(0.2x^5\) for \(\dfrac{1}{5}x^5\) and \(-4\sqrt{x^3}\) or \(-4\sqrt{x}^3\) or \(-4x\sqrt{x}\) or \(-4x^{1.5}\) for \(-4x^{\frac{3}{2}}\)
Do not allow \(-3x^1\) for this mark.
Condone spurious integral signs e.g. \(\displaystyle\int \dfrac{1}{5}x^5 - 4x^{\frac{3}{2}} - 3x + c\) or d\(x\) left in their answer
ISW after a correct expression seen e.g. if they multiply through by 5 or e.g. try to solve = 0
Do not allow e.g. \(\dfrac{1}{5}x^5 + -4x^{\frac{3}{2}} + -3x + c\)