| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2022 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find indefinite integral of polynomial/power |
| Difficulty | Easy -1.2 This is a straightforward application of standard integration rules for powers of x. Each term integrates independently using the power rule, requiring only recall of the formula and basic algebraic manipulation. No problem-solving or conceptual insight needed—purely routine AS-level integration practice. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x^n \to x^{n+1}\) | M1 | For raising any correct power of \(x\) by 1 including \(5 \to 5x\) (not for \(+ c\)). Also allow e.g. \(x^3 \to x^{3+1}\) |
| \(\int\left(8x^3 - \frac{3}{2\sqrt{x}} + 5\right)dx = \frac{8x^4}{4} \ldots + 5x\) | A1 | For 2 correct non-fractional power terms (allow unsimplified coefficients), may appear on separate lines. Indices must be processed. \(+c\) does not count as a correct term here. Condone the 1 appearing as a power or denominator such as \(\frac{5x^1}{1}\) |
| \(= \ldots -2 \times \frac{3}{2}x^{\frac{1}{2}} + \ldots\) | A1 | For the correct fractional power term (allow unsimplified). Allow e.g. \(+-2\times1.5\sqrt{x^1}\). Also allow fractions within fractions such as \(\dfrac{\frac{3}{2}}{\frac{1}{2}}x^{\frac{1}{2}}\) |
| \(= 2x^4 - 3x^{\frac{1}{2}} + 5x + c\) | A1 | All correct and simplified on one line including \(+c\). Allow \(-3\sqrt{x}\) or \(-\sqrt{9x}\) for \(-3x^{\frac{1}{2}}\). Do not accept \(+-3x^{\frac{1}{2}}\). Award once a correct expression is seen and isw, but if there is any additional/incorrect notation and no correct expression has been seen on its own, withhold the final mark. e.g. \(\int 2x^4 - 3x^{\frac{1}{2}} + 5x + c\, dx\) or \(2x^4 - 3x^{\frac{1}{2}} + 5x + c = 0\) with no correct expression seen earlier are both A0. |
| (4 marks) |
## Question 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^n \to x^{n+1}$ | M1 | For raising any correct power of $x$ by 1 including $5 \to 5x$ (not for $+ c$). Also allow e.g. $x^3 \to x^{3+1}$ |
| $\int\left(8x^3 - \frac{3}{2\sqrt{x}} + 5\right)dx = \frac{8x^4}{4} \ldots + 5x$ | A1 | For 2 correct non-fractional power terms (allow unsimplified coefficients), may appear on separate lines. Indices must be processed. $+c$ does not count as a correct term here. Condone the 1 appearing as a power or denominator such as $\frac{5x^1}{1}$ |
| $= \ldots -2 \times \frac{3}{2}x^{\frac{1}{2}} + \ldots$ | A1 | For the correct fractional power term (allow unsimplified). Allow e.g. $+-2\times1.5\sqrt{x^1}$. Also allow fractions within fractions such as $\dfrac{\frac{3}{2}}{\frac{1}{2}}x^{\frac{1}{2}}$ |
| $= 2x^4 - 3x^{\frac{1}{2}} + 5x + c$ | A1 | All correct and simplified on one line including $+c$. Allow $-3\sqrt{x}$ or $-\sqrt{9x}$ for $-3x^{\frac{1}{2}}$. Do not accept $+-3x^{\frac{1}{2}}$. Award once a correct expression is seen and isw, but if there is any additional/incorrect notation and no correct expression has been seen on its own, withhold the final mark. e.g. $\int 2x^4 - 3x^{\frac{1}{2}} + 5x + c\, dx$ or $2x^4 - 3x^{\frac{1}{2}} + 5x + c = 0$ with no correct expression seen earlier are both A0. |
| **(4 marks)** | | |\begin{enumerate}
\item Find
\end{enumerate}
$$\int \left( 8 x ^ { 3 } - \frac { 3 } { 2 \sqrt { x } } + 5 \right) \mathrm { d } x$$
giving your answer in simplest form.
\hfill \mbox{\textit{Edexcel AS Paper 1 2022 Q1 [4]}}