∫cscx dx

=∫ 1/sinx dx

=∫ 1/[2 sin(x/2)cos(x/2)]dx

=∫ 1/[sin(x/2)cos(x/2)]d(x/2)

=∫ 1/tan(x/2)* seconds? (x/2) d(x/2)

=∫ 1/tan(x/2)d[tan(x/2)](∫sec？ (x/2)d(x/2)=tan(x/2)+C)

=ln|tan(x/2)|+C

Extended data:

A definite integral is a number and an indefinite integral is an expression. They only have a mathematical relationship. A function can have indefinite integral without definite integral or definite integral without definite integral.

Continuous function must have definite integral and indefinite integral; If there are only finite discontinuous points on the finite interval [a, b] and the function is bounded, then the definite integral exists; If there are jumping points, going points and infinite discontinuous points, the original function must not exist, that is, the indefinite integral must not exist.