| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2008 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Definite integral with trigonometric functions |
| Difficulty | Standard +0.3 This is a structured two-part question where part (i) guides students to find constants by comparing coefficients (a routine algebraic technique), and part (ii) applies this result to evaluate a definite integral using reverse chain rule. The setup is scaffolded and the techniques are standard C4 material, making it slightly easier than average but still requiring multiple steps and careful execution. |
| Spec | 1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Perform an operation to produce an equation connecting A and B (or possibly in A or in B) | M1 | Probably substituting value of \(\theta\), or comparing coefficients of \(\sin x\), and/or \(\cos x\) |
| \(A = 2\) | A1 | 3 |
| \(B = -2\) | A1 | |
| (ii) Write \(4\sin\theta\) as \(A(\sin\theta + \cos\theta) + B(\cos\theta - \sin\theta)\) | A and B need not be numerical – but, if they are, they should be the values found in (i). | |
| and re-write integrand as \(A + \frac{B(\cos\theta - \sin\theta)}{\sin\theta + \cos\theta}\) | M1 | |
| \(\int A\,d\theta = A\theta\) | \(\sqrt{B1}\) | general or numerical |
| \(\int \frac{B(\cos\theta - \sin\theta)}{\sin\theta + \cos\theta}\,d\theta = B\ln(\sin\theta + \cos\theta)\) | \(\sqrt{A2}\) | general or numerical |
| Produce \(\frac{1}{4}A\pi + B\ln\sqrt{2}\) f.t. with their A,B | \(\sqrt{A1}\) | 5 |
(i) Perform an operation to produce an equation connecting A and B (or possibly in A or in B) | M1 | Probably substituting value of $\theta$, or comparing coefficients of $\sin x$, and/or $\cos x$ |
$A = 2$ | A1 | 3 | WW scores 3 |
$B = -2$ | A1 | |
(ii) Write $4\sin\theta$ as $A(\sin\theta + \cos\theta) + B(\cos\theta - \sin\theta)$ | | A and B need not be numerical – but, if they are, they should be the values found in (i). |
and re-write integrand as $A + \frac{B(\cos\theta - \sin\theta)}{\sin\theta + \cos\theta}$ | M1 | |
$\int A\,d\theta = A\theta$ | $\sqrt{B1}$ | general or numerical |
$\int \frac{B(\cos\theta - \sin\theta)}{\sin\theta + \cos\theta}\,d\theta = B\ln(\sin\theta + \cos\theta)$ | $\sqrt{A2}$ | general or numerical |
Produce $\frac{1}{4}A\pi + B\ln\sqrt{2}$ f.t. with their A,B | $\sqrt{A1}$ | 5 | Expect $\frac{1}{2}\pi - \ln 2$ (Numerical answer only) |(i) Given that
$$A ( \sin \theta + \cos \theta ) + B ( \cos \theta - \sin \theta ) \equiv 4 \sin \theta$$
find the values of the constants $A$ and $B$.\\
(ii) Hence find the exact value of
$$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 4 \sin \theta } { \sin \theta + \cos \theta } \mathrm { d } \theta$$
giving your answer in the form $a \pi - \ln b$.
\hfill \mbox{\textit{OCR C4 2008 Q7 [8]}}