Cos xdy + ysin xdx = dx Помогите решить

Перепишем:
$Cos xdy + ysin xdx = dx|*\frac{1}{dxcosx}\\y'+ytgx=\frac{1}{cosx}\\y=uv;y'=u'v+v'u\\u'v+v'u+uvtgx=\frac{1}{cosx}\\u'v+u(v'+vtgx)=\frac{1}{cosx}\\\begin{cases}v'+vtgx=0\\u'v=\frac{1}{cosx}\end{cases}\\\frac{dv}{dx}+vtgx=0|*\frac{dx}{v}\\\frac{dv}{v}=-tgxdx\\\int\frac{dv}{v}=-\int tgxdx\\ln|v|=ln|cosx|\\v=cosx\\\frac{ducosx}{dx}=\frac{1}{cosx}|*\frac{dx}{cosx}\\du=\frac{dx}{cos^2x}\\\int du=\int\frac{dx}{cos^2x}\\u=tgx+C\\y=cosx(tgx+C)=sinx+Ccosx$