Define the function $f(x)$ by $f(x) = \frac{2}{(2x - 1)(2x + 1)}$.
(a)[2]
Decompose $f(x)$ into partial fractions.
(b)[2]
Using the expression found in part (a), show that $$\bigl(f(x)\bigr)^2 = \frac{1}{(2x - 1)^2} - \frac{1}{2x - 1} + \frac{1}{2x + 1} + \frac{1}{(2x + 1)^2}.$$
(c)[5]
Hence, show that $\int_{1}^{2} (f(x))^2\,dx = \dfrac{2}{5} + \dfrac{1}{2} \ln\left(\dfrac{5}{9}\right)$.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “State or imply the form $\dfrac{A}{2x-1}+\dfrac{B}{2x+1}$ and apply an appropriate method to determine $A$ or $B$” …