| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Integrate after simplifying a quotient |
| Difficulty | Moderate -0.8 This is a straightforward P1 integration question requiring algebraic manipulation to split the fraction and rewrite terms with fractional powers before integrating term-by-term. The manipulation (dividing each term by 2√x to get 2x^(3/2) + (1/2)x^(-1/2)) and subsequent integration using the power rule are routine techniques with no problem-solving insight required, making it easier than average. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{4x^2+1}{2\sqrt{x}} = \frac{4x^2}{2\sqrt{x}} + \frac{1}{2\sqrt{x}} = 2x^{\frac{3}{2}} + \frac{1}{2}x^{-\frac{1}{2}}\) | M1 | Attempts to write \(\frac{4x^2+1}{2\sqrt{x}}\) as a sum of two terms. Award if any index is correct and processed for form \(Px^m + Qx^n\). Do not allow unprocessed indices |
| \(2x^{\frac{3}{2}} + \frac{1}{2}x^{-\frac{1}{2}}\) | A1 | oe. Coefficients do not have to be simplified. Condone \(\frac{4x^{\frac{3}{2}}+x^{-\frac{1}{2}}}{2}\) for M1A1 |
| \(\int \frac{4x^2+1}{2\sqrt{x}}\,dx = \frac{4}{5}x^{\frac{5}{2}} + x^{\frac{1}{2}} + c\) | M1 | Raises the power of one correct index by one. Index must be processed correctly, so award for sight of \(\to \ldots x^{\frac{5}{2}}\) or \(\ldots x^{\frac{1}{2}}\) |
| \(\frac{4}{5}x^{\frac{5}{2}} + x^{\frac{1}{2}} + c\) | A1 | One correct term in simplest form: either \(+\frac{4}{5}x^{\frac{5}{2}}\) or \(+x^{\frac{1}{2}}\) |
| \(\frac{4}{5}x^{\frac{5}{2}} + x^{\frac{1}{2}} + c\) | A1 | Fully correct. Accept exact equivalent simplified answers such as \(0.8x^2\sqrt{x}+\sqrt{x}+c\). Condone spurious notation such as \(\int\) or \(dx\). Condone \(\frac{4}{5}x^{\frac{5}{2}}+1x^{\frac{1}{2}}+c\) |
## Question 4:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{4x^2+1}{2\sqrt{x}} = \frac{4x^2}{2\sqrt{x}} + \frac{1}{2\sqrt{x}} = 2x^{\frac{3}{2}} + \frac{1}{2}x^{-\frac{1}{2}}$ | M1 | Attempts to write $\frac{4x^2+1}{2\sqrt{x}}$ as a sum of two terms. Award if any index is correct and processed for form $Px^m + Qx^n$. Do not allow unprocessed indices |
| $2x^{\frac{3}{2}} + \frac{1}{2}x^{-\frac{1}{2}}$ | A1 | oe. Coefficients do not have to be simplified. Condone $\frac{4x^{\frac{3}{2}}+x^{-\frac{1}{2}}}{2}$ for M1A1 |
| $\int \frac{4x^2+1}{2\sqrt{x}}\,dx = \frac{4}{5}x^{\frac{5}{2}} + x^{\frac{1}{2}} + c$ | M1 | Raises the power of one correct index by one. Index must be processed correctly, so award for sight of $\to \ldots x^{\frac{5}{2}}$ or $\ldots x^{\frac{1}{2}}$ |
| $\frac{4}{5}x^{\frac{5}{2}} + x^{\frac{1}{2}} + c$ | A1 | One correct term in simplest form: either $+\frac{4}{5}x^{\frac{5}{2}}$ or $+x^{\frac{1}{2}}$ |
| $\frac{4}{5}x^{\frac{5}{2}} + x^{\frac{1}{2}} + c$ | A1 | Fully correct. Accept exact equivalent simplified answers such as $0.8x^2\sqrt{x}+\sqrt{x}+c$. Condone spurious notation such as $\int$ or $dx$. Condone $\frac{4}{5}x^{\frac{5}{2}}+1x^{\frac{1}{2}}+c$ |
**(5 marks)**\begin{enumerate}
\item Find
\end{enumerate}
$$\int \frac { 4 x ^ { 2 } + 1 } { 2 \sqrt { x } } d x$$
giving the answer in its simplest form.
$$\int \frac { 4 x ^ { 2 } + 1 } { 2 \sqrt { x } } \mathrm {~d} x$$
giving the answer in its simplest form.
\hfill \mbox{\textit{Edexcel P1 2019 Q4 [5]}}