6 Show that \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } \left( 4 \cos ^ { 2 } 2 x + \frac { 1 } { \cos ^ { 2 } x } \right) \mathrm { d } x = \frac { 3 } { 4 } \sqrt { 3 } + \frac { 1 } { 6 } \pi - 1\).
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Question 6:
Answer Marks
Guidance
Answer Marks
Guidance
Express \(4\cos^2 2x\) in the form \(k_1\cos 4x + k_2\) M1
Where \(k_1k_2 \neq 0\)
Obtain correct \(2\cos 4x + 2\) A1
Allow unsimplified
State or imply \(\frac{1}{\cos^2 x} = \sec^2 x\) B1
Maybe implied by integration
Integrate to obtain \(k_3\sin 4x + k_4x + \tan x\) *M1
Where \(k_3k_4 \neq 0\)
Obtain correctly \(\frac{1}{2}\sin 4x + 2x + \tan x\) A1
Use limits correctly with correct values of \(\sin\frac{4}{3}\pi\) and \(\tan\frac{1}{3}\pi\) indicated DM1
Confirm given result \(\frac{3}{4}\sqrt{3} + \frac{1}{6}\pi - 1\) with sufficient detail A1
AG
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## Question 6:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Express $4\cos^2 2x$ in the form $k_1\cos 4x + k_2$ | M1 | Where $k_1k_2 \neq 0$ |
| Obtain correct $2\cos 4x + 2$ | A1 | Allow unsimplified |
| State or imply $\frac{1}{\cos^2 x} = \sec^2 x$ | B1 | Maybe implied by integration |
| Integrate to obtain $k_3\sin 4x + k_4x + \tan x$ | *M1 | Where $k_3k_4 \neq 0$ |
| Obtain correctly $\frac{1}{2}\sin 4x + 2x + \tan x$ | A1 | |
| Use limits correctly with correct values of $\sin\frac{4}{3}\pi$ and $\tan\frac{1}{3}\pi$ indicated | DM1 | |
| Confirm given result $\frac{3}{4}\sqrt{3} + \frac{1}{6}\pi - 1$ with sufficient detail | A1 | AG |
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6 Show that $\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } \left( 4 \cos ^ { 2 } 2 x + \frac { 1 } { \cos ^ { 2 } x } \right) \mathrm { d } x = \frac { 3 } { 4 } \sqrt { 3 } + \frac { 1 } { 6 } \pi - 1$.\\
\hfill \mbox{\textit{CAIE P2 2023 Q6 [7]}}