Evaluate the indefinite integral as an infinite series: integral of (cos(x) – 1)/x dx?

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!

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