This paper describes the CAI system for indefinite integral. The system has five fundamental functions for intelligent tutoring system as follows : 1. Interactive teaching, 2. Problem solving, 3. Error detection, diagnosis, and remedial action, 4. Explanation for WHY and WHAT questions, and 5. User-friendly interface by mouse and window system.
And students can use the system on 16-bit personal computer.
The system consists of the five parts such as “interface”, “input analysis”, “student model”, “pedagogical strategy”, and “knowledge base”. In the interface, there are four modes : input, request of hints, questions and termination modes. The student can select each mode according to his understanding. In the input analysis, a student who selected the input mode in the interface part inputs the solving procedure, e. g. “I solve the problem in an indefinite integral”, or “I use the distribution law”. Then he inputs a solution of the problem. The validity of the student's solving procedure is evaluated by comparison between his procedure and that created by the system. The validity of the student's solution is evaluated by comparison between his and solution according to the knowledge base of the system. Then the system diagnoses student's understanding based on the both validities. The student model consists of seven lists such as  list of student's understanding about solving procedure,  list of student's understanding about solution,  list of solving skills,  bug rules,  number of careless mistakes,  number of hints required by student and  documents of input of student and output of the system. In the pedagogical strategy, the system gives the student some hints and easier problems to get the initiative from him. The knowledge base is written in inference rules of the prolog style and is constructed hierarchically. Thirty-two of high school students solved the ten problems before/after use of the system. The results of the tests before/after showed 82.5 % of unacquired knowledge was aquired by use of the system.