Evaluate the indefinite integral as an infinite series:

**Answer 1**

Since

cos x = sum(k=0 to infinity) (-1)^k x^(2k)/(2k)!,

(cos x – 1)/x = [-1 + sum(k=0 to infinity) (-1)^k x^(2k)/(2k)!] / x

= sum(k=1 to infinity) (-1)^k x^(2k-1)/(2k)!

Integrating term by term yields

int (cos x – 1) dx /x = C + sum(k=1 to infinity) (-1)^k x^(2k)/[(2k)(2k)!].

I hope that helps!