CAIE P3 2015 November — Question 5 8 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2015
SessionNovember
Marks8
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Mark schemeDownload PDF ↗
TopicIntegration by Substitution
TypeSubstitution u = expression involving trig (non-pure sin/cos)
DifficultyStandard +0.8 This is a moderately challenging integration by substitution question requiring multiple steps: finding du/dx, expressing sin(2x) in terms of cos(x), then substituting to eliminate x entirely. The algebraic manipulation of sin(2x) = 2sin(x)cos(x) and relating cos(x) back to u adds complexity beyond routine substitution. However, the substitution is given explicitly and the technique is standard for P3 level, preventing it from being truly difficult.
Spec1.05a Sine, cosine, tangent: definitions for all arguments1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08h Integration by substitution
5 Use the substitution \(u = 4 - 3 \cos x\) to find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 9 \sin 2 x } { \sqrt { ( 4 - 3 \cos x ) } } \mathrm { d } x\).
AnswerMarks Guidance
State \(du = 3\sin x \, dx\) or equivalentB1
Use identity \(\sin 2x = 2\sin x \cos x\)B1
Carry out complete substitution, for \(x\) and \(dx\)M1
Obtain \(\int\frac{8 - 2u}{\sqrt{u}} \, du\) or equivalentA1
Integrate to obtain expression of form \(au^{\frac{1}{2}} + bu^{\frac{3}{2}}\), \(ab \neq 0\)M1*
Obtain correct \(16u^{\frac{1}{2}} - \frac{4}{3}u^{\frac{3}{2}}\)A1
Apply correct limits correctlydep M1*
Obtain \(\frac{20}{3}\) or exact equivalentA1 [8] 
State $du = 3\sin x \, dx$ or equivalent | B1 |
Use identity $\sin 2x = 2\sin x \cos x$ | B1 |
Carry out complete substitution, for $x$ and $dx$ | M1 |
Obtain $\int\frac{8 - 2u}{\sqrt{u}} \, du$ or equivalent | A1 |
Integrate to obtain expression of form $au^{\frac{1}{2}} + bu^{\frac{3}{2}}$, $ab \neq 0$ | M1* |
Obtain correct $16u^{\frac{1}{2}} - \frac{4}{3}u^{\frac{3}{2}}$ | A1 |
Apply correct limits correctly | dep M1* |
Obtain $\frac{20}{3}$ or exact equivalent | A1 | [8]
5 Use the substitution $u = 4 - 3 \cos x$ to find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 9 \sin 2 x } { \sqrt { ( 4 - 3 \cos x ) } } \mathrm { d } x$.

\hfill \mbox{\textit{CAIE P3 2015 Q5 [8]}}
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