A beautiful integral feat. an infinite series result

This video evaluates $\int_{0}^{\infty} \cos x /(e^{x}+e^{-x})\,dx$ by two independent routes. The first expands the integrand as a geometric series in $e^{-2x}$, exchanges summation and integration, and identifies each term as the Laplace transform of $\cos x$ evaluated at $2k+1$, yielding the series $\sum_{k=0}^{\infty} (-1)^k(2k+1)/[(2k+1)^{2}+1]$ and yes I know I forgot the (-1)^k term in the video (see pinned comment). The second exploits the integrand's even symmetry, invokes a couple of my favorite tricks in special functions and results in a lovely closed form giving both the integral and infinite series results. Reference on justifying the interchange of summation and integration operators:    • FINALLY understanding the switch up of ope...   My complex analysis lectures:    • Complex Analysis Lectures   If you like the videos and would like to support the channel:   / maths505   You can follow me on Instagram for write ups that come in handy for my videos and DM me in case you need math help: https://instagram.com/maths.505?igshi... My LinkedIn:   / kamaal-mirza-86b380252   Advanced MathWear: https://my-store-ef6c0f.creator-sprin...