| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Double integration by parts |
| Difficulty | Standard +0.3 This is a standard double integration by parts question requiring systematic application of the technique twice, followed by evaluation of a definite integral. While it requires more steps than single integration by parts, it follows a well-practiced algorithmic procedure with no conceptual surprises, making it slightly easier than average for C4 level. |
| Spec | 1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Integration by parts applied: \(x^2e^x - \int \lambda xe^x \{dx\}\), \(\lambda > 0\) | M1 | Must be in this form |
| \(x^2e^x - \int 2xe^x \{dx\}\) | A1 oe | |
| Achieving \(\pm Ax^2e^x \pm Bxe^x \pm C\int e^x\{dx\}\) or \(\pm K\int xe^x\{dx\} \rightarrow \pm K\left(xe^x - \int e^x\{dx\}\right)\) | M1 | \(A\neq 0\), \(B\neq 0\), \(C\neq 0\); can be implied |
| \(\pm Ax^2e^x \pm Bxe^x \pm Ce^x\) | M1 | \(A\neq 0\), \(B\neq 0\), \(C\neq 0\) |
| \(x^2e^x - 2(xe^x - e^x) \ \{+c\}\) i.e. \(x^2e^x - 2xe^x + 2e^x \ \{+c\}\) | A1 | Correct answer with/without \(+c\) |
| [5 marks] |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Applies limits of 1 and 0 to expression of form \(\pm Ax^2e^x \pm Bxe^x \pm Ce^x\), \(A\neq 0\), \(B\neq 0\), \(C\neq 0\), and subtracts correct way round | M1 | Evidence of proper consideration of limit of 0 required; just subtracting zero is M0 |
| \(e - 2\) | A1 cso | Do not allow \(e - 2e^0\) unless simplified; 0.718... without seeing \(e-2\) is A0; correct solution only |
| [2 marks] |
## Question 1:
### Part (a): $\int x^2 e^x \, dx$
| Working/Answer | Mark | Guidance |
|---|---|---|
| Integration by parts applied: $x^2e^x - \int \lambda xe^x \{dx\}$, $\lambda > 0$ | M1 | Must be in this form |
| $x^2e^x - \int 2xe^x \{dx\}$ | A1 oe | |
| Achieving $\pm Ax^2e^x \pm Bxe^x \pm C\int e^x\{dx\}$ **or** $\pm K\int xe^x\{dx\} \rightarrow \pm K\left(xe^x - \int e^x\{dx\}\right)$ | M1 | $A\neq 0$, $B\neq 0$, $C\neq 0$; can be implied |
| $\pm Ax^2e^x \pm Bxe^x \pm Ce^x$ | M1 | $A\neq 0$, $B\neq 0$, $C\neq 0$ |
| $x^2e^x - 2(xe^x - e^x) \ \{+c\}$ i.e. $x^2e^x - 2xe^x + 2e^x \ \{+c\}$ | A1 | Correct answer with/without $+c$ |
| **[5 marks]** | | |
### Part (b): $\left[x^2e^x - 2(xe^x - e^x)\right]_0^1$
| Working/Answer | Mark | Guidance |
|---|---|---|
| Applies limits of 1 and 0 to expression of form $\pm Ax^2e^x \pm Bxe^x \pm Ce^x$, $A\neq 0$, $B\neq 0$, $C\neq 0$, and subtracts correct way round | M1 | Evidence of proper consideration of limit of 0 required; just subtracting zero is M0 |
| $e - 2$ | A1 cso | Do not allow $e - 2e^0$ unless simplified; 0.718... without seeing $e-2$ is A0; correct solution only |
| **[2 marks]** | | |
**Total: [7 marks]**\begin{enumerate}[label=(\alph*)]
\item Find $\int x ^ { 2 } e ^ { x } d x$.
\item Hence find the exact value of $\int _ { 0 } ^ { 1 } x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 2013 Q1 [7]}}