Q.Evaluate: $\int_0^\pi \dfrac{\sin 3x\,(1+2\cos x)}{(1+\cos x)^2}\,dx$

Honest content gap. The source extraction for this item reads "sin3x (1 + 2 cos x) (1 + cos x) 2 dx" with no visible fraction bar, so the exact grouping of numerator versus denominator, and whether "sin3x" means $\sin(3x)$ or $\sin^3x$, cannot be confirmed from the extracted text alone. Testing the most natural reading — $\dfrac{\sin3x\,(1+2\cos x)}{(1+\cos x)^2}$ — near the upper limit $x=\pi$ shows $1+\cos x\sim\varepsilon^2/2$ (so the denominator $\sim\varepsilon^4$) while the numerator only vanishes to order $\varepsilon$ (or $\varepsilon^3$ under the $\sin^3x$ reading), so the integrand blows up and the definite integral over $[0,\pi]$ diverges under every gr …

Always check the integrand stays bounded across the whole interval, including the endpoi …

The failure mode this avoids: silently picking a plausible grouping and reporting a confident-looking closed …