Coroneos: 100 Integrals Worked Solutions

Elias never solved the hundredth integral. For forty years, he had filled notebooks with curled ribbons of symbols – ∫, dx, the patient grammar of calculus. Coroneos' list was his bible. Numbers 1 through 99 fell like dominoes. But number 100 – ∫ √(tan x) dx – refused.

∫ x sin(x) dx → Integration by parts: let u = x, dv = sin x dx → du = dx, v = -cos x = -x cos x + ∫ cos x dx = -x cos x + sin x + C coroneos 100 integrals worked solutions

∫ ln x dx → Integration by parts: u = ln x, dv = dx → du = (1/x) dx, v = x = x ln x - ∫ dx = x ln x - x + C Elias never solved the hundredth integral

That night, he wrote the substitution u = √(tan x). The page sighed. Then he let the pen rest. Some solutions are not answers. They are permission to stop. He closed the notebook and went to make tea. Please clarify which one you actually want (or both separately), and I’ll give a full response. Numbers 1 through 99 fell like dominoes

"No," Elias whispered. "It's a door."

∫ e^x cos x dx → Integration by parts twice (or use complex exponentials): result = (e^x (sin x + cos x))/2 + C