f`(x)=(1/sinx)*(sinx)`=cosx/sinx=ctgx
L= ∫ ^(π/2)_(π/3)sqrt(1+(ctgx)^2) dx= ∫ ^(π/2)_(π/3)sqrt(1/sin^2x) dx=∫ ^(π/2)_(π/3)(1/sinx) dx=ln|tg(x/2)|)||^(π/2)_(π/3)=
=ln|tg(π/4)|-ln|tg(π/6)|=ln1-ln(1/sqrt(3))=0-ln3^(-1/2))=(1/2)ln3
О т в е т. (1/2)ln3