Theorem 1: If f(x) is continuous in the interval [a, b], then f(x) is integrable in [a, b].

Theorem 2: If the interval f(x) is bounded on [a, b] and there are only finite discontinuous points, then f(x) is integrable on [a, b].

Theorem 3: Let f(x) be monotone in the interval [a, b], then f(x) can be integrated in [a, b].

If f(x) is a continuous function on [a, b] and f ′ (x) = f (x), then the value of the definite integral formula is the difference between the value of the original function at the upper limit and the value of the original function at the lower limit.

It is precisely because of this theory that the relationship between integral and Riemann integral is revealed, which shows its important position in calculus and even higher mathematics. Therefore, Newton-Leibniz formula is also called the basic theorem of calculus.