OCR C4 2010 January — Question 3 5 marks

Exam BoardOCR
ModuleC4 (Core Mathematics 4)
Year2010
SessionJanuary
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStandard Integrals and Reverse Chain Rule
TypeUse trig identity before definite integration
DifficultyModerate -0.3 This question requires knowing the double angle formula cos(2x) = 2cos²(x) - 1, then simplifying the integrand to 2 - sec²(x), which integrates to standard forms. While it involves multiple steps (identity substitution, simplification, integration, evaluation), each step is routine for C4 level. The main challenge is recognizing which identity to use, but this is explicitly prompted. Slightly easier than average due to the guided approach and standard techniques.
Spec1.05l Double angle formulae: and compound angle formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)

3 By expressing \(\cos 2 x\) in terms of \(\cos x\), find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } \frac { \cos 2 x } { \cos ^ { 2 } x } \mathrm {~d} x\).

3 By expressing $\cos 2 x$ in terms of $\cos x$, find the exact value of $\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } \frac { \cos 2 x } { \cos ^ { 2 } x } \mathrm {~d} x$.

\hfill \mbox{\textit{OCR C4 2010 Q3 [5]}}