Can we integrate exactly this problem.

Question

$$\int \frac{\cos x}{2x}dx$$ when $\cos x$ is a finite differential family and $x^{-1}$ is a infinite differential family then how we can solve this.

Answer 1

Like $\frac{\sin x}{x},$ this has no elementary antiderivative. The definite integral can be evaluated by using numerical methods, and the integral from 0 to $\infty$ has nice solution.

You can also evaluate the integral as a Taylor series. Use the fact that $\cos x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}.$