grandes-ecoles 2019 Q19

grandes-ecoles · France · centrale-maths2__pc Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts

For every natural integer $n$ and every real $x$, set $I_n(x) = \int_0^{\pi/2} \cos(2xt)(\cos t)^n \, \mathrm{d}t$. Show $$\forall n \in \llbracket 2, +\infty\llbracket, \forall x \in \mathbb{R}, \quad \left(1 - \frac{4x^2}{n^2}\right) I_n(x) = \frac{n-1}{n} I_{n-2}(x) \quad \text{and} \quad \left(1 - \frac{4x^2}{n^2}\right) \frac{I_n(x)}{I_n(0)} = \frac{I_{n-2}(x)}{I_{n-2}(0)}.$$

For every natural integer $n$ and every real $x$, set $I_n(x) = \int_0^{\pi/2} \cos(2xt)(\cos t)^n \, \mathrm{d}t$. Show
$$\forall n \in \llbracket 2, +\infty\llbracket, \forall x \in \mathbb{R}, \quad \left(1 - \frac{4x^2}{n^2}\right) I_n(x) = \frac{n-1}{n} I_{n-2}(x) \quad \text{and} \quad \left(1 - \frac{4x^2}{n^2}\right) \frac{I_n(x)}{I_n(0)} = \frac{I_{n-2}(x)}{I_{n-2}(0)}.$$

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