Show that ∫⁴₁ (2/x + 2/(2x+1)) dx = ln 48.

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2019). The mark or mark type is shown in brackets.

1. [M1] Integrate to obtain form k₁ ln x + k₂ ln(2x + 1)
2. [A1] Obtain correct 2ln x + ln(2x + 1)
3. [M1] Use logarithm addition/subtraction property correctly
4. [M1] Use logarithm power property correctly
5. [A1] Confirm ln 48 with no errors seen
6. [B1] Use identity sin 2x = 2 sin x cos x
7. [B1] State or imply cot x + 2 cosec x = cos x / sin x + 2 / sin x
8. [M1] Attempt to express integrand in terms of cos 2x and cos x
9. [A1] Obtain correct integrand 1 + cos 2x + 4 cos x
10. [M1] Integrate to obtain at least terms k₃ sin 2x and k₄ sin x
11. [A1] Obtain correct x + ½ sin 2x + 4 sin x + c