Definite integral with special limits

A question is this type if and only if the reduction formula involves definite integrals with limits that produce specific numerical values (like powers of 2, 3, or expressions involving √2, √3) in the recurrence relation.

1 questions · Challenging +1.3

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AQA Further Paper 2 2024 June Q20
9 marks Challenging +1.3
The integral \(I_n\) is defined by $$I_n = \int_0^{\frac{\pi}{4}} \cos^n x \, dx \quad\quad (n \geq 0)$$
  1. Show that $$I_n = \left(\frac{n-1}{n}\right)I_{n-2} + \frac{1}{n\left(2^{\frac{n}{2}}\right)} \quad\quad (n \geq 2)$$ [6 marks]
  2. Use the result from part (a) to show that $$\int_0^{\frac{\pi}{4}} \cos^6 x \, dx = \frac{a\pi + b}{192}$$ where \(a\) and \(b\) are integers to be found. [3 marks]