Lời giải:
\(\int ^{\frac{\pi}{2}}_{0}\frac{\sin 2x\cos x}{1+\cos x}dx=\int ^{\frac{\pi}{2}}_{0}\frac{2\sin x\cos ^2x}{\cos x+1}dx=2\int ^{\frac{\pi}{2}}_{0}\frac{\cos^2x\sin xdx}{\cos x+1}\)
\(=2\int ^{\frac{\pi}{2}}_{0}\frac{-\cos ^2xd(\cos x)}{\cos x+1}=2\int ^{0}_{1}\frac{-t^2dt}{t+1}=2\int ^{1}_{0}\frac{t^2}{t+1}dt\)
\(=2\int^1_0\frac{(t^2-1)+1}{t+1}dt=2\int ^1_0(t-1+\frac{1}{t+1})dt\)
\(=2(\frac{t^2}{2}-t+\ln|t+1|)|^{1}_0=2\ln 2-1\)
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