Question

Solve: $\int \frac{\cos x}{x}dx$

Answer 1

$\int \frac{cosx}{x}dx$ can't be done in elementary

functions.

But we can use Taylor series

$\Rightarrow cosx=1-\frac{k^2}{2!}+\frac{k^4}{4!}-\frac{k^6}{6!}+---$

$\Rightarrow \frac{cosx}{x}=\frac{1}{x}-\frac{k}{2!}+\frac{k^3}{4!}-\frac{k^5}{6!}+---$

$\Rightarrow \int \frac{cosx}{k}dx=lnx-\frac{x^2}{2.2!}+\frac{k^4}{4.4!}-\frac{k^6}{6.6!}+----+C$