tg(x/2)=t
тогда
(x/2)=arctgt
x=2arctgt
dx=2dt/(1+t^2)
sinx=2t/(1+t^2)
∫ dx/(5+4sinx)= ∫ (2dt/(1+t^2))/ (5+(8t)/(1+t2))=
= ∫ 2dt/(5t^2+8t+5)=
выделяем полный квадрат
[5t^2+8t+5=5(t^2+(8/5)t+1)=5·(t+(4/5))^2–(16/25)+1)]
=(2/5) ∫ dt/(t+(4/5))^2+(9/25))=
=(2/5)·(1/(3/5))arctg (t+(4/5))/(3/5)+C=
=(2/3)arctg ((5tg(x/2)+4)/3)+C