f(x)= d x ^6 x is a polynomial of degree

Step-by-step solution

Let f(x)= d x ^6 x f(x)= cosec ^6 x d x From reduction formula, we have aligned & I _ n = cosec ^n x d x & =- cosec ^ n-2 x x n-1 + n-2 n-1 I _ n-2 aligned gathered f(x)=- cosec ^4 x x 5 + 4 5 [ - cosec ^2 x x 3 + 2 3 I ₂ ] =- cosec ^4 x x 5 - 4 15 cosec ^2 x x+ 8 15 [- x] = - (1+ ^2 x )^2 x 5 - 4 15 (1+ ^2 x ) x = -1 5 [1+ ^4 x+2 ^2 x ] x- 4 15 [- x) ( cosec ^2 x=1+ ^2 x ) = -1 5 [ x+ ^3 x ] -4 15 ^3 x- 4 15 ^3 x- 8 15 x x] gathered aligned & = -15 15 x- ^5 x 5 - 10 15 ^3 x & = - ^5 x 5 - 2 3 ^3 x- x aligned It is