Mathematical analysis discusses integral divergence of 200

How can it be so troublesome as it is said upstairs!

e (sinx) is a positive bounded quantity. So just think about it.

sin2x/x^pdx。

Consider whether it is absolutely convergent:

sin2x bounded; Rule.

x=0 is the defective integral, p-1<1 converges, and p>0, i.e., 0 considers positive infinity, and p>1 converges.

So 1 As for the convergence of conditions, come back and write again, and go to class.

If the conditions converge, it is more difficult to consider directly; However, when 1p>=2, x=0 is flawed and does not converge;

So it is only necessary to discuss 00 <>

The method of judging the divergence of the generalized integral is that it is a fixed value calculated after integration, which is either infinity or convergence; After integralization, the calculation is not a fixed value, it is infinity, or divergence. The generalized integral discriminant method only needs to study the properties of the integrand function itself, and its divergence can be known.

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).

The generalized integral discriminant method is not only more refined than the traditional discriminant method, but also avoids the difficulty of the traditional discriminant method in finding a reference function.

The integral intervals of definite integrals are finite, and the integrand is 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.

Apply the limit form of Cauchy's discriminant.

Let l=lim(x->+x p [x a*(lnx) b]=lim(x->+x (p-a)] [(lnx) b](1) let p>1

When a>=p>1, l=0, so the original integral converges.

2) Let p<=1

When a1, the original integral = [1 (1-b)]*1 (lnx) (b-1)|(3,+=1 (b-1)(ln3) (b-1), convergence.

In summary, when a>1, the original integral converges.

At 01, the original integral converges.

The part where the denominator is 0lim(x->1+) 1 [ x-1)(3-x)]0 is 1 (x-1).

Likewise. The part of lim(x->3-) 1 [ x-1)(3-x)]0 is 1 (3-x).

The divergence of integrals is mainly in the following cases:

1) One of the upper and lower limits of the integrals, or at the same time tends to infinity;

2) The integrand tends to infinity at one or more points in the integral region.

To examine the convergence of the integral, you can find the limit after the integration to see if the limit exists: existence is convergence; If it doesn't exist, it diverges.

For an integral such as 1 (x-a) p, a is a point within the integration region and can be judged to converge according to the magnitude of the p value: p < 1; Divergence in other cases.

Direct calculation (or definition).

That is, the divergence is judged by directly calculating the anomalous integral. If the anomalous integral can calculate a specific value, it converges, otherwise it diverges. This method is suitable for the discriminant discrepancy of the convergence of the anomalous integral when the original function of the integrand function is easy to find.

Compare the limit forms of the convergence method.

The ordinary form of the comparative discriminant method is relatively simple, so I will not repeat it, and then I will summarize the limit form of the comparative discriminant method.

In general, the divergence of generalized integrals can be judged as follows:1If the specific value can be found by integrals, then of course it means that it is convergent; If it is found that it tends to infinity according to the calculation of the definite integral, then of course it means that it is divergent; 2.

If you can't figure out the exact value, you can scale it by inequality, here.

The judgment of the convergence of defective integrals is one of the difficulties of students' learning, and the methods of judging the convergence of defective integrals mainly include the definition method, the comparative method, the Cauchy discriminant method, the Dirichlet discriminant method and the Abelian discriminant method. The original function of the integrand function is known or easily found by the definition method; Functions that satisfy the conditions of the Dirichlet discriminant are distinguished by the Dirichlet discriminant; The function that satisfies the conditions of the Abelian discriminant is used by the Abelian discriminant; Bounded functions such as sine and cosine or functions with absolute convergence can be judged by comparative method.

The method of judging the divergence of the generalized integral is that it is a fixed value calculated after integration, which is either infinity or convergence; After the integration, the calculation is not a fixed value, it is infinity, or the divergence of the observer. The generalized integral discriminant method can know its divergence as long as the properties of the lead number itself are studied.

In general, the divergence of generalized integrals can be judged in this way:

1.If the specific value can be found by integrals, then of course it means that it is convergent; If it is found that it tends to infinity according to the calculation of the definite integral, then of course it means that it is divergent;

2.If it is not easy to calculate the specific value, it can be scaled through the inequality, and there are too many specific situations here to repeat it.

Then the following two topics can be analyzed like this:

1.Its indefinite integral can be found, you might as well find the indefinite integral 2The indefinite integral can be found but is not continuous at 3, but does not affect the substitution calculation in the specific steps