Now, I've tried a couple of different substitutions and integrating partially but unfortunately, to no luck, I was wondering about your thoughts on it. I'd also be very thankful if someone were to have a complete answer: $$\int {\cos x(2\sin x+3)\over (\sin^2 x +2\sin x+3) \sqrt{\sin^2x+2\sin x+4}}\,dx.$$ Thanks in advance. :)
2026-04-12 20:54:01.1776027241
Find $\int {\cos x(2\sin x+3)\over (\sin^2 x +2\sin x+3) \sqrt{\sin^2x+2\sin x+4}}\,dx$
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HINT: $$ \int {\cos x(2\sin x+3)\over (\sin^2 x +2\sin x+3) \sqrt{\sin^2x+2\sin x+4}}\,dx =\\ \int {\cos x(2\sin x+2)\over (\sin^2 x +2\sin x+3) \sqrt{\sin^2x+2\sin x+4}}\,dx \\ +\int\frac{\cos x}{(\sin^2 x +2\sin x+3) \sqrt{\sin^2x+2\sin x+4}}\,dx $$ For the first term you can use substitution $u=\sin^2 x+2\sin x+3$, for the second one: $t=\sin x$, which leads to rather easy integrals.