| Exam Board | Edexcel |
|---|---|
| Module | P4 (Pure Mathematics 4) |
| Year | 2023 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Substitution then integration by parts |
| Difficulty | Standard +0.8 This P4 question requires executing a guided substitution to simplify a complex integrand, then applying integration by parts twice to evaluate ∫u²eᵘ du. While the substitution is scaffolded in part (a), students must handle algebraic manipulation of surds, change limits correctly, and perform repeated integration by parts—a multi-step process requiring careful bookkeeping. This is moderately challenging but follows standard P4 techniques without requiring novel insight. |
| Spec | 1.08h Integration by substitution1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(a=3, b=5\) | B1 | Both values seen in solution; may be recovered in (b) |
| \(u=\sqrt{2x+1} \Rightarrow \frac{dx}{du}=u\) or \(\frac{du}{dx}=(2x+1)^{-\frac{1}{2}}\) o.e. | B1 | Correct expression involving \(\frac{du}{dx}\) or \(\frac{dx}{du}\) or \(du\) and \(dx\) separately; may be unsimplified |
| \(\int\sqrt{8x+4}\,e^{\sqrt{2x+1}}dx = \int 2u\,e^u\,u\,du\) | M1 | Attempts to fully change integral to one with respect to \(u\); must include attempt at replacing \(dx\) |
| \(= \int_3^5 2u^2 e^u\,du\) | A1 | Complete method with correct limits and \(du\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int 2u^2 e^u\,du = 2u^2e^u - \int 4ue^u\,du\) | M1 | Integration by parts once to obtain \(pu^2e^u - \int que^u\,du\), where \(p,q>0\) |
| \(= 2u^2e^u-(4ue^u-4e^u) = 2u^2e^u-4ue^u+4e^u\) | dM1 A1ft | dM1: Completely integrates by parts twice to form \(pu^2e^u-que^u\pm re^u\); A1ft: \(ku^2e^u-2kue^u+2ke^u\) accepted |
| \(\int_4^{12}\sqrt{8x+4}\,e^{\sqrt{2x+1}}dx = \left[2u^2e^u-4ue^u+4e^u\right]_3^5 = (50e^5-20e^5+4e^5)-(18e^3-12e^3+4e^3)\) | ddM1 | Substitutes \(a\) and \(b\) into form \(pu^2e^u-que^u\pm re^u\) and subtracts; must use a value for \(k\) |
| \(= 34e^5-10e^3\) | A1 | \(34e^5-10e^3\) or exact equivalent e.g. \(2e^3(17e^2-5)\) |
## Question 4(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $a=3, b=5$ | **B1** | Both values seen in solution; may be recovered in (b) |
| $u=\sqrt{2x+1} \Rightarrow \frac{dx}{du}=u$ or $\frac{du}{dx}=(2x+1)^{-\frac{1}{2}}$ o.e. | **B1** | Correct expression involving $\frac{du}{dx}$ or $\frac{dx}{du}$ or $du$ and $dx$ separately; may be unsimplified |
| $\int\sqrt{8x+4}\,e^{\sqrt{2x+1}}dx = \int 2u\,e^u\,u\,du$ | **M1** | Attempts to fully change integral to one with respect to $u$; must include attempt at replacing $dx$ |
| $= \int_3^5 2u^2 e^u\,du$ | **A1** | Complete method with correct limits and $du$ |
---
## Question 4(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int 2u^2 e^u\,du = 2u^2e^u - \int 4ue^u\,du$ | **M1** | Integration by parts once to obtain $pu^2e^u - \int que^u\,du$, where $p,q>0$ |
| $= 2u^2e^u-(4ue^u-4e^u) = 2u^2e^u-4ue^u+4e^u$ | **dM1 A1ft** | dM1: Completely integrates by parts twice to form $pu^2e^u-que^u\pm re^u$; A1ft: $ku^2e^u-2kue^u+2ke^u$ accepted |
| $\int_4^{12}\sqrt{8x+4}\,e^{\sqrt{2x+1}}dx = \left[2u^2e^u-4ue^u+4e^u\right]_3^5 = (50e^5-20e^5+4e^5)-(18e^3-12e^3+4e^3)$ | **ddM1** | Substitutes $a$ and $b$ into form $pu^2e^u-que^u\pm re^u$ and subtracts; must use a value for $k$ |
| $= 34e^5-10e^3$ | **A1** | $34e^5-10e^3$ or exact equivalent e.g. $2e^3(17e^2-5)$ |
---\begin{enumerate}
\item (a) Using the substitution $u = \sqrt { 2 x + 1 }$, show that
\end{enumerate}
$$\int _ { 4 } ^ { 12 } \sqrt { 8 x + 4 } \mathrm { e } ^ { \sqrt { 2 x + 1 } } \mathrm {~d} x$$
may be expressed in the form
$$\int _ { a } ^ { b } k u ^ { 2 } \mathrm { e } ^ { u } \mathrm {~d} u$$
where $a$, $b$ and $k$ are constants to be found.\\
(b) Hence find, by algebraic integration, the exact value of
$$\int _ { 4 } ^ { 12 } \sqrt { 8 x + 4 } e ^ { \sqrt { 2 x + 1 } } d x$$
giving your answer in simplest form.
\hfill \mbox{\textit{Edexcel P4 2023 Q4 [9]}}