1, this formula can be simply described as "the integral of even power function on a symmetric interval is equal to the even power of the function value at the midpoint of the interval divided by the corresponding odd power". The modern form of this formula is as follows: If f(x) is integrable on (a, b) and f(x) is an even function, then ∫f(x)dx=2∫f(x)dx is on (0, b).
2. The proof of this formula is simple. First of all, we pay attention to the symmetry of even function, that is, f(-x)=f(x). Then, we use this property to rearrange the integrals of the interval (a, b) and get ∫f(x)dx=∫f(x)dx+∫f(-x)dx. We notice that the second integral can be converted into -∫f(-x)dx.
3. Then we notice that the first integral on the interval (0, b) is equal to twice the integral on the interval (0, b/2), because the result of the integral on (0, b/2) is the same as that on (b/2, b). We notice that the second integral can also be converted into a double integral in the interval (0, b/2) (also because of symmetry).
The meaning of definite integral
1, definite integral is an important concept in mathematical analysis, which involves the integral operation of a function in a certain interval. The definite integral can be expressed as the limit of the sum of the integral of the function on the interval and the function value at the end of the interval.
2. There are many methods to calculate definite integral, among which the most common methods are rectangle method, trapezoid method and Simpson method. These methods are all based on the definition of definite integral, that is, the weighted average value of function value is used to approximate the integral value of function in a certain interval.
3. Definite integral is widely used. It can not only help us solve the integral of function, but also solve the problems of area, volume, average value and extreme value. For example, we can use definite integral to solve the volume of the rotating body, the length of the curve, the arc length of the curve, the area of the plane area and so on.