OCR FP2 2011 January — Question 1 5 marks

Exam BoardOCR
ModuleFP2 (Further Pure Mathematics 2)
Year2011
SessionJanuary
Marks5
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Mark schemeDownload PDF ↗
TopicIntegration by Substitution
TypeIndefinite integral with Weierstrass or trigonometric substitution
DifficultyChallenging +1.2 This is a standard Weierstrass substitution (t-formulae) problem from Further Maths FP2. While the substitution itself is given, students must recall the t-formulae for sin x and cos x, handle the algebraic manipulation of the resulting rational function, and integrate correctly. It's harder than typical A-level integration due to the Further Maths context and multi-step algebraic complexity, but it's a textbook application of a well-known technique rather than requiring novel insight.
Spec4.08h Integration: inverse trig/hyperbolic substitutions
1 Use the substitution \(t = \tan \frac { 1 } { 2 } x\) to find \(\int \frac { 1 } { 1 + \sin x + \cos x } \mathrm {~d} x\).
Question 1:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(t = \tan\frac{1}{2}x \Rightarrow dt = \frac{1}{2}\sec^2\frac{1}{2}x\,dx = \frac{1}{2}(1+t^2)\,dx\)B1 For correct result AEF (may be implied)
\(\int\frac{1}{1+\sin x+\cos x}\,dx = \int\frac{1}{1+\frac{2t}{1+t^2}+\frac{1-t^2}{1+t^2}}\cdot\frac{2}{1+t^2}\,dt\)M1 For substituting throughout for \(x\)
A1For correct unsimplified \(t\) integral
\(= \int\frac{1}{1+t}\,dt = \ln1+t (+c)\)
\(= \ln k\left1+\tan\frac{1}{2}x\right (+c)\) 
## Question 1:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $t = \tan\frac{1}{2}x \Rightarrow dt = \frac{1}{2}\sec^2\frac{1}{2}x\,dx = \frac{1}{2}(1+t^2)\,dx$ | B1 | For correct result **AEF** (may be implied) |
| $\int\frac{1}{1+\sin x+\cos x}\,dx = \int\frac{1}{1+\frac{2t}{1+t^2}+\frac{1-t^2}{1+t^2}}\cdot\frac{2}{1+t^2}\,dt$ | M1 | For substituting throughout for $x$ |
| | A1 | For correct unsimplified $t$ integral |
| $= \int\frac{1}{1+t}\,dt = \ln|1+t|(+c)$ | M1 | For integrating (even incorrectly) to $a\ln|f(t)|$. Allow $|\,|$ or $(\,)$ |
| $= \ln k\left|1+\tan\frac{1}{2}x\right|(+c)$ | A1 **5** | For correct $x$ expression; $k$ may be numerical, $c$ is not required |

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1 Use the substitution $t = \tan \frac { 1 } { 2 } x$ to find $\int \frac { 1 } { 1 + \sin x + \cos x } \mathrm {~d} x$.

\hfill \mbox{\textit{OCR FP2 2011 Q1 [5]}}
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