Tgx/2=t⇒x=2arctgt .dx=2dt/(1+t²)
sinx=2t/(1+t²),cosx=(1-t²)/(1+t²)
∫dx/(4cosx+3sinx=∫2dt/(4-4t²+6t)=-1/2*∫dt/(t²-3t-1)=-1/2*∫dt/[(t-3/4)²-(5/4)²]=
-1/2*2/5*ln|(t-3/4-5/4)/(t-3/4+5/4)|=-1/5*ln|(t-2)/(t-1/2)|=
=-1/5*ln|(tgx/2-2)/(tgx/2+1/2)|+C=-1/5*ln|(tgx/2-2)/(tgx/2+1/2)|+C