Question 7 - A-Level Maths

Exam Board	CAIE
Module	P3 (Pure Mathematics 3)
Year	2020
Session	June
Topic	Integration with Partial Fractions

Express $\mathrm { f } ( x )$ in partial fractions.
Using your answer to part (a), show that $$( f ( x ) ) ^ { 2 } = \frac { 1 } { ( 2 x - 1 ) ^ { 2 } } - \frac { 1 } { 2 x - 1 } + \frac { 1 } { 2 x + 1 } + \frac { 1 } { ( 2 x + 1 ) ^ { 2 } }$$
Hence show that $\int _ { 1 } ^ { 2 } ( \mathrm { f } ( x ) ) ^ { 2 } \mathrm {~d} x = \frac { 2 } { 5 } + \frac { 1 } { 2 } \ln \left( \frac { 5 } { 9 } \right)$.