Задача 34969 ...
Условие
математика ВУЗ
955
Все решения
tg(x/2)=t
x/2=arctgt
x=2arctgt
dx=2/(1+t^2)
sinx=2t/(1+t^2)
cosx=(1-t^2)/(1+t^2)
3 + sinx + cosx = 3 + 2t/(1+t^2) + (1-t^2)/(1+t^2)
3 + sinx + cosx = (3+3t^2+2t+1-t^2)/(1+t^2)
∫ dx/(3 + sinx + cosx)= ∫ 2dt/(2t^2+2t+4)=
= ∫ dt/(t^2+t+2)= ∫ dt/((t+(1/2))^2+(7/4))=
=(1/sqrt(7/4))*arctg(t+(1/2))/sqrt(7/2)+C=
=(2/sqrt(7))* arctg((2t+1)/sqrt(7)) + C=
= [b](2/sqrt(7)) * arctg ((2tg(x/2)+1)/sqrt(7))+C[/b]
