OCR C4 2009 January — Question 4 6 marks

Exam BoardOCR
ModuleC4 (Core Mathematics 4)
Year2009
SessionJanuary
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStandard Integrals and Reverse Chain Rule
TypeDefinite integral with trigonometric functions
DifficultyModerate -0.3 This question requires expanding the bracket to get 1 + 2sin(x) + sin²(x), using the identity sin²(x) = (1-cos(2x))/2, then integrating standard forms. While it involves multiple steps and a trigonometric identity, these are routine C4 techniques with straightforward execution and no problem-solving insight required, making it slightly easier than average.
Spec1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08d Evaluate definite integrals: between limits
4 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( 1 + \sin x ) ^ { 2 } \mathrm {~d} x\).
Question 4:
AnswerMarks Guidance
Answer/WorkingMark Guidance
Attempt to expand \((1+\sin x)^2\) and integrate it\*M1 Minimum of \(1+\sin^2 x\)
Attempt to change \(\sin^2 x\) into \(f(\cos 2x)\)M1
Use \(\sin^2 x = \frac{1}{2}(1-\cos 2x)\)A1 dep M1+M1
Use \(\int \cos 2x\, dx = \frac{1}{2}\sin 2x\)A1 dep M1+M1
Use limits correctly on an attempt at integrationdep\*M1 Tolerate \(g\left(\frac{1}{4}\pi\right)-0\)
\(\frac{3}{8}\pi - \sqrt{2} + \frac{7}{4}\) AE(3-term)FA1 WW \(1.51\ldots \to\) M1 A0
Total: 6 
# Question 4:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt to expand $(1+\sin x)^2$ and integrate it | \*M1 | Minimum of $1+\sin^2 x$ |
| Attempt to change $\sin^2 x$ into $f(\cos 2x)$ | M1 | |
| Use $\sin^2 x = \frac{1}{2}(1-\cos 2x)$ | A1 | dep M1+M1 |
| Use $\int \cos 2x\, dx = \frac{1}{2}\sin 2x$ | A1 | dep M1+M1 |
| Use limits correctly on an attempt at integration | dep\*M1 | Tolerate $g\left(\frac{1}{4}\pi\right)-0$ |
| $\frac{3}{8}\pi - \sqrt{2} + \frac{7}{4}$ AE(3-term)F | A1 | WW $1.51\ldots \to$ M1 A0 |
| **Total: 6** | | |

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4 Find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( 1 + \sin x ) ^ { 2 } \mathrm {~d} x$.

\hfill \mbox{\textit{OCR C4 2009 Q4 [6]}}
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