| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2020 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Show substitution transforms integral, then apply integration by parts or further substitution |
| Difficulty | Standard +0.8 This question requires numerical approximation with rectangles, then a non-trivial substitution followed by integration by parts (the 2t sin(t/2) integral). While substitution itself is standard A-level, the combination of techniques and the need to handle the composite function through multiple transformations makes this moderately challenging, though still within typical Further Maths scope. |
| Spec | 1.08h Integration by substitution1.08i Integration by parts1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.25\{\sin 0 + \sin(\frac{1}{2}\sqrt{0.25}) + \sin(\frac{1}{2}\sqrt{0.5}) + \sin(\frac{1}{2}\sqrt{0.75})\}\) | M1 | Attempt four rectangles of width 0.25, with height on left-hand side; no need to see \(\sin 0\); allow M1 if evaluated in degrees (0.00452) |
| Lower bound \(= 0.253\) | A1 | Obtain 0.253 or better; soi as lower bound |
| \(0.25(\sin(\frac{1}{2}\sqrt{0.25}) + \sin(\frac{1}{2}\sqrt{0.5}) + \sin(\frac{1}{2}\sqrt{0.75}) + \sin\frac{1}{2})\) | M1 | Attempt rectangles of width 0.25 with height on right-hand side; or subtract \(\sin 0\) from part (ii) and add \(\sin 0.5\); M0 if \(\sin 0\) explicitly included |
| Upper bound \(= 0.373\) or \(0.374\) | A1 | Obtain 0.373 or 0.374 (from rounding upper bound up), or better; soi as upper bound |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(t^2 = x - 1\) | M1 | Attempt to link \(dt\) and \(dx\) |
| \(2t\,dt = dx\) | A1 | Obtain correct equation linking \(dt\) and \(dx\); allow \(dt = \frac{1}{2}(x-1)^{-\frac{1}{2}}dx\) |
| \(\int \sin(\frac{1}{2}\sqrt{x-1})\,dx = \int \sin(\frac{1}{2}t)\,2t\,dt = \int 2t\sin(\frac{1}{2}t)\,dt\) A.G. | A1 | Attempt integrand in terms of \(t\) to obtain given answer; award A1 once all elements correct, even if not in same order; BOD if no brackets |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(-4t\cos(\frac{1}{2}t) + \int 4\cos(\frac{1}{2}t)\,dt\) | M1* | Attempt integration by parts; correct parts; as far as first stage |
| \(-4t\cos(\frac{1}{2}t) + 8\sin(\frac{1}{2}t)\) | A1 | Correct integral |
| \((-4\cos\frac{1}{2} + 8\sin\frac{1}{2}) - (-0 + 0)\) | M1d* | Attempt use of limits; using either \(t\) or \(x\), but must be consistent; condone no clear use of lower limit for M1 |
| \(8\sin\frac{1}{2} - 4\cos\frac{1}{2}\) AG | A1 | Obtain given answer; must see some indication that lower limit considered |
| [4] |
## Question 10 - Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.25\{\sin 0 + \sin(\frac{1}{2}\sqrt{0.25}) + \sin(\frac{1}{2}\sqrt{0.5}) + \sin(\frac{1}{2}\sqrt{0.75})\}$ | M1 | Attempt four rectangles of width 0.25, with height on left-hand side; no need to see $\sin 0$; allow M1 if evaluated in degrees (0.00452) |
| Lower bound $= 0.253$ | A1 | Obtain 0.253 or better; soi as lower bound |
| $0.25(\sin(\frac{1}{2}\sqrt{0.25}) + \sin(\frac{1}{2}\sqrt{0.5}) + \sin(\frac{1}{2}\sqrt{0.75}) + \sin\frac{1}{2})$ | M1 | Attempt rectangles of width 0.25 with height on right-hand side; or subtract $\sin 0$ from part (ii) and add $\sin 0.5$; M0 if $\sin 0$ explicitly included |
| Upper bound $= 0.373$ or $0.374$ | A1 | Obtain 0.373 or 0.374 (from rounding upper bound up), or better; soi as upper bound |
| | **[4]** | |
## Question 10 - Part (b)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $t^2 = x - 1$ | M1 | Attempt to link $dt$ and $dx$ |
| $2t\,dt = dx$ | A1 | Obtain correct equation linking $dt$ and $dx$; allow $dt = \frac{1}{2}(x-1)^{-\frac{1}{2}}dx$ |
| $\int \sin(\frac{1}{2}\sqrt{x-1})\,dx = \int \sin(\frac{1}{2}t)\,2t\,dt = \int 2t\sin(\frac{1}{2}t)\,dt$ **A.G.** | A1 | Attempt integrand in terms of $t$ to obtain given answer; award A1 once all elements correct, even if not in same order; BOD if no brackets |
| | **[3]** | |
## Question 10 - Part (b)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $-4t\cos(\frac{1}{2}t) + \int 4\cos(\frac{1}{2}t)\,dt$ | M1* | Attempt integration by parts; correct parts; as far as first stage |
| $-4t\cos(\frac{1}{2}t) + 8\sin(\frac{1}{2}t)$ | A1 | Correct integral |
| $(-4\cos\frac{1}{2} + 8\sin\frac{1}{2}) - (-0 + 0)$ | M1d* | Attempt use of limits; using either $t$ or $x$, but must be consistent; condone no clear use of lower limit for M1 |
| $8\sin\frac{1}{2} - 4\cos\frac{1}{2}$ **AG** | A1 | Obtain given answer; must see some indication that lower limit considered |
| | **[4]** | |
---10\\
\includegraphics[max width=\textwidth, alt={}, center]{febe231d-200a-4957-b41b-de5b9be98b0a-7_352_545_258_239}
The diagram shows the curve $y = \sin \left( \frac { 1 } { 2 } \sqrt { x - 1 } \right)$, for $1 \leqslant x \leqslant 2$.
\begin{enumerate}[label=(\alph*)]
\item Use rectangles of width 0.25 to find upper and lower bounds for $\int _ { 1 } ^ { 2 } \sin \left( \frac { 1 } { 2 } \sqrt { x - 1 } \right) \mathrm { d } x$. Give your answers correct to 3 significant figures.
\item \begin{enumerate}[label=(\roman*)]
\item Use the substitution $t = \sqrt { x - 1 }$ to show that $\int \sin \left( \frac { 1 } { 2 } \sqrt { x - 1 } \right) \mathrm { d } x = \int 2 t \sin \left( \frac { 1 } { 2 } t \right) \mathrm { d } t$.
\item Hence show that $\int _ { 1 } ^ { 2 } \sin \left( \frac { 1 } { 2 } \sqrt { x - 1 } \right) \mathrm { d } x = 8 \sin \frac { 1 } { 2 } - 4 \cos \frac { 1 } { 2 }$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR H240/01 2020 Q10 [11]}}