Standard +0.8 This is a moderately challenging integration question requiring substitution with trigonometric identities, careful manipulation of the resulting integral (likely requiring double angle formulas or reduction), and evaluation of definite integral bounds. The 9-mark allocation and requirement for exact form indicate substantial working beyond routine substitution, placing it above average difficulty but not at the level requiring novel geometric insight.
Use the substitution \(2x = \tan \theta\) to find the exact value of
$$\int_0^{\frac{1}{2}} \frac{12}{(1 + 4x^2)^2} \, dx .$$
Give your answer in the form \(a + b\pi\), where \(a\) and \(b\) are rational numbers. [9]
Use the substitution $2x = \tan \theta$ to find the exact value of
$$\int_0^{\frac{1}{2}} \frac{12}{(1 + 4x^2)^2} \, dx .$$
Give your answer in the form $a + b\pi$, where $a$ and $b$ are rational numbers. [9]
\hfill \mbox{\textit{CAIE P3 2024 Q11 [9]}}