[m]2cos^2x+3sin^2x=cos^2x(2+3 tg^2x)[/m]
[m]d(tgx)=\frac{dx}{cos^2x}[/m]
[m]∫ \frac{dx}{2cos^2x+3sin^2x}= ∫ \frac{dx}{cos^x(2+3tg^2x)}= ∫ \frac{d(tgx)}{2+3tg^2x}=[/m]
[m]tgx=t[/m]
[m]∫ \frac{dt}{2+3t^2}=\frac{1}{3}∫ \frac{dt}{\frac{2}{3}+t^2}=[/m] табличный интеграл
[m]=\frac{1}{3}\cdot \frac{1}{\sqrt{\frac{2}{3}}} arctg \frac{t}{\sqrt{\frac{2}{3}}}+C=\frac{1}{\sqrt{6}}arctg\frac{\sqrt{3}t}{\sqrt{2}}+C=\frac{1}{\sqrt{6}}arctg\frac{\sqrt{3}tgx}{\sqrt{2}}+C[/m]