CAIE P3 2014 June — Question 2 5 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2014
SessionJune
Marks5
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Mark schemeDownload PDF ↗
TopicIntegration by Substitution
TypeSubstitution u = expression involving trig (non-pure sin/cos)
DifficultyStandard +0.3 This is a straightforward substitution problem where the substitution is given explicitly. Students need to find du = 3sec²x dx, change limits (u: 1→4), and recognize the integral becomes ∫u^(1/2)/3 du. The algebra is clean and the final integration is routine, making this slightly easier than average despite involving trigonometry.
Spec1.08h Integration by substitution
2 Use the substitution \(u = 1 + 3 \tan x\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sqrt { } ( 1 + 3 \tan x ) } { \cos ^ { 2 } x } d x$$
AnswerMarks Guidance
State \(\frac{du}{dx} = 3\sec^2 x\) or equivalentB1
Express integral in terms of \(u\) and \(du\) (accept unsimplified and without limits)M1
Obtain \(\int \frac{1}{3}u^3 \, du\)A1
Integrate \(Cu^3\) to obtain \(\frac{2C}{3}u^3\)M1
Obtain \(\frac{14}{9}\)A1 [5] 
State $\frac{du}{dx} = 3\sec^2 x$ or equivalent | B1 |
Express integral in terms of $u$ and $du$ (accept unsimplified and without limits) | M1 |
Obtain $\int \frac{1}{3}u^3 \, du$ | A1 |
Integrate $Cu^3$ to obtain $\frac{2C}{3}u^3$ | M1 |
Obtain $\frac{14}{9}$ | A1 | [5]
2 Use the substitution $u = 1 + 3 \tan x$ to find the exact value of

$$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sqrt { } ( 1 + 3 \tan x ) } { \cos ^ { 2 } x } d x$$

\hfill \mbox{\textit{CAIE P3 2014 Q2 [5]}}
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Q1 5 Q2 5 Q3 6 Q4 6 Q5 8 Q6 8 Q7 8 Q8 9 Q9 10 Q10 10