## Problems on Improper Integrals Exercise 1. Decide on the convergence or divergence of Answer. First notice that the denominator is equal to 0 when x=1. Then the function inside the integral sign is unbounded at x=1. Hence we have two bad points 1 and . So we must split the integral and write Let us first take care of the integral We have Since (by Bertrand's test) the improper integral is convergent, then by the comparison test the improper integral Next we take care of the integral First notice that when , then One may check this by showing that On the other hand (and this is the crucial step in this exercise) is to find a polynomial approximation of when . This will be done via Taylor polynomials. Indeed, we have when , which gives Hence we have which implies when . The p-test implies that the improper integral is convergent. Therefore the limit test implies that the improper integral is convergent. Putting the two integrals together, we conclude that the improper integral is convergent.

Exercise 2. Decide on the convergence or divergence of Answer. The term is never equal for . So let us focus on the term . According to the domain of the tangent function, the only bad points we have to worry about is . Clearly we have Hence we have an improper integral or Type I at the bad point . Clearly we have when . So let us approximate when . Again we will use Taylor polynomials, we have when . This gives when . Putting the stuff together we get The p-test implies that the improper integral is divergent. Therefore the improper integral is divergent.

Exercise 3. Decide on the convergence or divergence of Answer. The only improper behavior is around . Hence this integral is of Type II not of Type I. Therefore no need for splitting it. Note that when , then . Hence when . The p-test implies that the improper integral is convergent. Therefore the limit test implies that the improper integral is convergent.

## Problems

Next you will find some not so easy problems on improper integrals. We invite you to solve them and submit the answer to SOS MATHematics. We will publish your answer with your name. Good luck.

Problem 1. First decide on the convergence and divergence of Then evaluate f(x).

Problem 2. Assume that is continuous. Find , where Problem 3. Consider the function Find f(x).

Problem 4. Evaluate Problem 5. In this problem, we will evaluate 1
Evaluate 2
Let which function (that is f is differentiable and f' is continuous). Show that 3
Evaluate Problem 6. Decide on convergence or divergence of where x > 0 and is any real number.

Problem 7. Decide on convergence or divergence of Problem 8. Decide on convergence or divergence of If I is convergent, evaluate it.

Problem 9. Find  [Geometry] [Algebra] [Differential Equations]
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Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard. Mohamed A. Khamsi
Tue Dec 3 17:39:00 MST 1996