A-LevelMathematicsIntegration9709/32 · May/June 2018 · Q48 Marks

Show that (2 sin x - sin 2x) / (1 - cos 2x) = sin x / (1 + cos x)

(i)Show that (2 sin x - sin 2x) / (1 - cos 2x) = sin x / (1 + cos x)[4]
(ii)Hence, showing all necessary working, find integral from π/3 to π/2 of (2 sin x - sin 2x) / (1 - cos 2x) dx, giving your answer in the form ln k.[4]

Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

[M1] Use correct double angle formulae and express LHS in terms of cos x and sin x
[A1] Obtain a correct expression
[M1] Complete method to get correct denominator e.g. by factorising to remove a factor of 1 - cos x
[A1] Obtain the given RHS correctly
OR (working R to L):
[M1A1] sin x/(1+cos x) * (1-cos x)/(1-cos x) = (sin x - sin x cos x) / (1-cos^2 x)
[M1A1] = (2 sin x - 2 sin x cos x) / (2 - 2 cos^2 x) = (2 sin x - sin 2x) / (1 - cos 2x)
[M1*] State integral of the form a ln(1+cos x)
[A1] Obtain integral - ln(1+cos x)
[M1(dep)*] Substitute correct limits in correct order
[A1] Obtain answer ln(3/2) or equivalent

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