Question

${\int }_0^{\pi /2}\frac{\sin x\cos x}{{\cos }^{2}x+3\cos x+2}dx=?$

Answer 1

${\int }_0^{\pi /2}\frac{\sin x\cos x}{{\cos }^{2}x+2\frac{3}{2}\cos x+\frac{9}{4}-\frac{9}{4}+2}dx$

${\int }_0^{\pi /2}\frac{\sin x\cos x}{(\cos x+\frac{3}{2}{)}^2-\frac{1}{4}}dx$

Putting $\cos x+\frac{3}{2}=m$

$-\sin x=dm$

${\int }_0^1\frac{m-\frac{3}{2}}{m^2-\frac{1}{4}}dm$

${\int }_0^1\frac{m-\frac{1}{2}-1}{m^2-\frac{1}{4}}dm$

${\int }_0^1\frac{1}{m+\frac{1}{2}}dm-{\int }_0^1\frac{1}{m^2-\frac{1}{4}}dm$

$\log |m+\frac{1}{2}|-{\int }_0^1\frac{1}{m-\frac{1}{2}}dm+{\int }_0^1\frac{1}{m+\frac{1}{2}}dm$

$\log \frac{|m+\frac{1}{2}{|}^2}{|m-\frac{1}{2}|}$

$\log \frac{9/4}{1/2}=\log \frac{9}{2}$