1. The main methods to find definite integral are substitution integral method and integration by parts. There are two substitution methods of definite integral. The first is differential integration, such as xdx= 1/2dx? , integral variable or x, just put x? Looking at a whole, the integral limit remains unchanged.

2. The second substitution integration method makes x=x(t), and naturally dx=dx(t)=x'(t)dt. A new variable is introduced here, and the integral limit should be changed from the transformation range of x to the variation range of t.

3. Partial integration, let u=u(x), v=v(x) all be derivable on the interval [a, b], and u ′, v ′ ∈ r ([a, b]), then there is a partial integration formula.

Second, the definition of interpretation

1 and typical problems solved by definite integral

(1) area of curved trapezoid

(2) Distance of variable speed linear motion

2. Sufficient conditions of function integrability

Theorem Let f(x) be continuous in the interval [a, b], then f(x) is integrable in the interval [a, b], that is, continuous => integrable.

Theorem Let f(x) be bounded on the interval [a, b] and have only a finite number of discontinuous points, then f(x) can be integrated on the interval [a, b].

3. Some important properties of definite integral

① Properties If f (x) in the interval [a, b] is ≥ 0, then ∫ ABF (x) dx is ≥ 0.

It is inferred that if f(x)≤g(x) on the interval [a, b], then ∫abf(x)dx≤∫abg(x)dx.

Inference |∫abf(x)dx|≤∫ab|f(x)|dx.

② Let m and m be the maximum and minimum values of the function f(x) in the interval [a, b] respectively, then m (b-a) ≤∫ ABF (x) dx ≤ m (b-a) shows that the approximate range of the integral value can be estimated by the maximum and minimum values of the integrand function in the integral interval.

Property (mean value theorem of definite integral) If the function f(x) is continuous in the interval [a, b], there is at least one point ξ in the interval [a, b], which makes the following formula hold: ∫ ABF (x) dx = f (ξ) (b-a).