If-I-n-cos-nx-cos-x-dx-then-1-n-

Question Number 152112 by peter frank last updated on 25/Aug/21

If I_{n} = \int \frac{\cos nx}{\cos x} dx then 1_{n} = ?

Answered by Olaf_Thorendsen last updated on 26/Aug/21

I_{n} = \int \frac{\cos (n x)}{\cos x} d x

I_{n} = \int \frac{T_{n} (\cos x)}{\cos x} d x

T_{n} : Tchebychev polynomial

T_{n} (X) = \sum_{0 ⩽ 2 k ⩽ n} C_{2 k}^{n} {(- 1)}^{k} X^{n - 2 k} {(1 - X^{2})}^{k}

We must distinguish two cases

n even and n odd \dots