Question

Решить интеграл dx/(2sinx+sin2x)

Answer 1

$\large \int\frac{dx}{2sinx+sin2x}=\frac{1}{2}\int\frac{dx}{sinx(1+cosx)}=\int\frac{(1+t^2)^2}{4(1+t^2)t}dt=\\\\=\frac{1}{4}\int\frac{1+t^2}{t}=\frac{1}{4}\int(\frac{1}{t}+t)dt=\frac{1}{4}ln|t|+\frac{t^2}{8}+C=\\\\=\frac{1}{4}ln|tg\frac{x}{2}|+\frac{tg^2\frac{x}{2}}{8}+C\\\\\\\\t=tg\frac{x}{2}\\x=2arctgt\\dx=\frac{2dt}{1+t^2}\\sinx=\frac{2t}{1+t^2}\ ;1+cosx=1+\frac{1-t^2}{1+t^2}=\frac{2}{1+t^2}$