Challenging +1.8 This is a challenging Further Maths question requiring derivation of a reduction formula through differentiation of a given expression, followed by recursive application. The hint guides the approach, but students must recognize how to manipulate the derivative to isolate the integral relationship, handle the boundary terms correctly, and then apply the formula multiple times with careful arithmetic. The multi-step nature, algebraic manipulation, and recursive calculation elevate this above standard integration by parts exercises.
The integral \(I_n\) is defined by
$$I_n = \int_0^{\frac{1}{4}\pi} \sec^n x \, dx.$$
By considering \(\frac{d}{dx}(\tan x \sec^n x)\), or otherwise, show that
$$(n + 1)I_{n+2} = 2^{\frac{4n}{n}} + nI_n.$$ [4]
Find the value of \(I_6\). [4]
The integral $I_n$ is defined by
$$I_n = \int_0^{\frac{1}{4}\pi} \sec^n x \, dx.$$
By considering $\frac{d}{dx}(\tan x \sec^n x)$, or otherwise, show that
$$(n + 1)I_{n+2} = 2^{\frac{4n}{n}} + nI_n.$$ [4]
Find the value of $I_6$. [4]
\hfill \mbox{\textit{CAIE FP1 2003 Q5 [8]}}