A-LevelMathematicsIntegration9709/22 · Oct/Nov 2018 · Q611 Marks

Show that ∫(from 1 to 6) (12 / (3x + 2)) dx = ln 256.

(a)Show that ∫(from 1 to 6) (12 / (3x + 2)) dx = ln 256.[5]
(b)Find the exact value of ∫(from 0 to π/6) (8 sin²x + tan² 2x) dx, showing all necessary working.[6]

Mark scheme

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

*M1 Integrate to obtain form kln(3x + 2)
A1 Obtain correct 4ln(3x + 2)
M1 Substitute limits correctly
M1 Apply relevant logarithm properties correctly
A1 Obtain ln 256 nfww
Condone poor use of brackets if recovered later
Dependent *M, must see kln20 – kln5 oe
Dependent *M, do not allow 4ln20 / 4ln5 oe, must be using both the subtraction and power laws correctly
AG; necessary detail needed
B1 Use identity to obtain 4(1-cos2x) oe
B1 Use identity to obtain sec² 2x-1
*M1 Integrate to obtain form k₁x + k₂ sin 2x + k₃ tan 2x
A1 Obtain correct 3x - 2sin2x + 1/2 tan2x
M1 Apply limits correctly retaining exactness
A1 Obtain π/3 - √3/2 or exact equivalent
Allow M1 if integrand contains pcos2x+qsec² 2x and no other trig terms
Dependent *M, allow sin(π/3), tan(π/3)

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