4 Find \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } x \sec ^ { 2 } x \mathrm {~d} x\). Give your answer in a simplified exact form.
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Question 4:
Answer Marks
Integrate by parts and reach \(ax\tan x + b\int\tan x\,dx\) M1*
Obtain \(x\tan x - \int\tan x\,dx\) A1
Complete the integration, obtaining a term \(\pm\ln\cos x\), or equivalent M1
Obtain integral \(x\tan x + \ln\cos x\), or equivalent A1
Substitute limits correctly, having integrated twice DM1
Use a law of logarithms M1
Obtain answer \(\frac{5}{18}\sqrt{3}\pi - \frac{1}{2}\ln 3\), or exact simplified equivalent A1
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## Question 4:
| Integrate by parts and reach $ax\tan x + b\int\tan x\,dx$ | M1* | |
| Obtain $x\tan x - \int\tan x\,dx$ | A1 | |
| Complete the integration, obtaining a term $\pm\ln\cos x$, or equivalent | M1 | |
| Obtain integral $x\tan x + \ln\cos x$, or equivalent | A1 | |
| Substitute limits correctly, having integrated twice | DM1 | |
| Use a law of logarithms | M1 | |
| Obtain answer $\frac{5}{18}\sqrt{3}\pi - \frac{1}{2}\ln 3$, or exact simplified equivalent | A1 | |
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4 Find $\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } x \sec ^ { 2 } x \mathrm {~d} x$. Give your answer in a simplified exact form.\\
\hfill \mbox{\textit{CAIE P3 2020 Q4 [7]}}