The indefinite integral of 1 and inx is ∫ ln xdx = xlnx-∫ xdlnx = xlnx-∫ dx = xlnx-x+c In calculus, the indefinite integral of function f, or the original function, or the inverse derivative, is a function f whose derivative is equal to f, that is, f' = f.

2. The relationship between indefinite integral and definite integral is determined by the basic theorem of calculus. Where f is the indefinite integral of f, according to Newton-Leibniz formula, the definite integral of many functions can be calculated simply by solving the indefinite integral. Here, we should pay attention to the relationship between indefinite integral and definite integral: definite integral is a number, while indefinite integral is an expression, and they have only one 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.