How to calculate anomalous integrals, and how to calculate anomalous integrals?

An anomalous integral with an integral limit of infinity may not be possible (but the integrand can be done if it has a flawed point), and it can be integrated with a sign and then converted into a numeric value with double.

The anomalous integral, also known as the generalized integral, is a generalization of the ordinary definite integral, which refers to the integral with the lower bound of the infinite upper limit, or the integrand with the imperfection point, the former is called the infinite generalized integral, and the latter is called the flawed integral (also known as the anomalous integral of the unbounded function).

Reply to li20061542's post Try quadgk, abnormal points, and flawed points are OK.

Reply to zsy312's post Oh, thank you, this also provides an idea, you can transform the integrand, replace it with a flawed integral, I'll try it.

Reply to rocwoods's post Okay, I'll check how to use it, thanks for the pointers!

The integral intervals of <> definite integrals are finite, and the integrands are bounded. However, in practical application and theoretical research, there will be some functions defined on infinite intervals or unbounded functions on finite intervals, and problems similar to definite integrals need to be considered for them.

Therefore, it is necessary to generalize the concept of definite integrals so that it can be applied to the above two types of functions. This kind of generalized integral, because it is different from the usual definite integral, is called a generalized integral, also called an anomalous integral.

For infinity of the upper and lower bounds, or for multiple imperfections in the integrated function, or a mixture of the above two types, it is called a mixed anomalous integral. For mixed anomalous integrals, multiple integration intervals must be split so that the original integral is the sum of two separate anomalous integrals, the infinite interval and the unbounded function.

There is the following algorithm for anomalous integrals:

Anomalous integration means that the derivative function of the calculated function has abnormal properties in (0,+, that is, it does not conform to the conventional integration method, and if the conditions of this integration are satisfied, the original function will be solved, then the anomalous integration method is used to find the original function.

When the value of the independent variable is negative after the derivative, a "" is added after the derivative, a situation is called an anomalous integral. When the value of the independent variable is negative, the original function has an anomaly property in (0,+, that is, when the value of the independent variable becomes a non-negative number after the derivative, then the original function has a solution to it.

If so, then write down the results and you're good to go. It is used for verification. However, the calculation process is cumbersome.

Let's move on to the calculation process later! Problem: When the independent variable after the derivative is positive, the integral is as follows (the integral form is unchanged) Solution:

This problem is relatively simple, because it is a continuous curve, so take the distance from a point on the x-axis to the straight line, i.e., y=x+y2. Then divide the curve into several segments to calculate.

Once you've done it, you'll find out that it's a credit! It's a simple question, but I don't think you know! So that's where we need to do it with the anomalous integral method!

The so-called anomalous integration method is a situation where the independent variable is zero and the derivative is non-negative. We need to find a little bit x0 (0).

The methods for calculating anomalous integrals are:

Theorem 1: Let f(x) be continuous over the interval [a,b], then f(x) is integrable over [a,b].

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

Theorem 3: Let f(x) be monotonic over the interval [a,b], then f(x) is integrable over [a,b].

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

Because of this theory, the connection between the integral and the essence of Riemann's accumulation of filial piety is revealed, which shows its important position in calculus and even higher mathematics of frank and prudent equilibrium, so the Newton-Leibniz formula is also called the fundamental theorem of calculus.

Normalcy integralsThe integral limit is finite, and the integrand is bounded (bounded is not necessarily integrable, integrable must be bounded).Anomalous PointsIf the limit of is present, then the anomalous integral converges.

Infinite anomalous integrals should have only one integral limit to infinity, and if the upper and lower limits are infinite, the integral should be disassembled.

By definition, using the limit to calculate.

Using a form similar to the Newton-Leibniz formula, if is a primitive function of , the writing is introduced.

Thus there is. Discuss the divergence of this anomalous integral.

If there is no boundary in any neighborhood, it is called a defect of .

In continuous, is the imperfection point, and in the anomalous integral is defined as.

Discuss the convergence of the integrals below.

Using the step-by-step integration method, you can put e (-x) [1+e (-x)] 2 behind dx and it will suddenly become clear.

First of all, the partial integration is obtained from the original function, and then the limit is solved.

When p=1, = 1 (xlnx)dx=lnlnxx +, limlnlnx=+ and the function diverges when p≠1.

1/(1-p)d(lnx)^(1-p)=1/(1-p)*(lnx)^(1-p)|(e, +x +, if 01,lim(lnx) (1-p)=0, then the original formula = 0-1 (1-p)=1 (p-1).