To evaluate the indefinite integral ∫ cos(x)/x dx as an infinite series, we start by recalling the Taylor series expansion for the cosine function:
cos(x) = ∑ (-1)n * (x)2n / (2n)! for n=0 to infinity.
Substituting this series into the integral gives:
∫ cos(x)/x dx = ∫ (∑ (-1)n * (x)2n / (2n)!)/x dx
This can be rewritten as:
∫ (∑ (-1)n * (x)2n-1 / (2n)!) dx
By interchanging the integral and the summation (justified by the uniform convergence of the series on any closed interval in the domain of integration), we have:
∑ (-1)n / (2n)! ∫ x2n-1 dx
Computing the integral:
∫ x2n-1 dx = (1/(2n)) * x2n + C
Thus, our series expression for the integral becomes:
∑ (-1)n * (x2n)/(2n * (2n)!) + C
This provides an infinite series representation for the indefinite integral of cos(x)/x. Therefore, we can conclude that:
∫ (cos(x)/x) dx = ∑ -1n * (x2n)/(2n * (2n)!) + C.