CAIE P3 2022 November — Question 3 5 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2022
SessionNovember
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIntegration by Parts
TypeBasic integration by parts
DifficultyStandard +0.3 This is a straightforward single application of integration by parts with standard functions (x and sec²x), where sec²x integrates to tan x. The definite integral evaluation is clean with the upper limit π/4 giving tan(π/4)=1. While it requires correct technique, it's a textbook example with no conceptual challenges beyond the standard method.
Spec1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.08i Integration by parts
3 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } x \sec ^ { 2 } x \mathrm {~d} x\).
Question 3:
AnswerMarks Guidance
AnswerMark Guidance
Commence integration by parts and reach \(x\tan x \pm \int \tan x \cdot 1\, dx\)*M1
Use a correct method to integrate \(\tan x\)M1
Obtain integral \(x\tan x - \ln\sec x\), or equivalentA1
Use limits correctly, having integrated twiceDM1
Obtain answer \(\frac{1}{4}\pi - \frac{1}{2}\ln 2\), or exact equivalentA1 
## Question 3:

| Answer | Mark | Guidance |
|--------|------|----------|
| Commence integration by parts and reach $x\tan x \pm \int \tan x \cdot 1\, dx$ | *M1 | |
| Use a correct method to integrate $\tan x$ | M1 | |
| Obtain integral $x\tan x - \ln\sec x$, or equivalent | A1 | |
| Use limits correctly, having integrated twice | DM1 | |
| Obtain answer $\frac{1}{4}\pi - \frac{1}{2}\ln 2$, or exact equivalent | A1 | |

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3 Find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } x \sec ^ { 2 } x \mathrm {~d} x$.\\

\hfill \mbox{\textit{CAIE P3 2022 Q3 [5]}}
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