MATH 1310.03F A Exam 2 Thursday, November 9, 2000

NAME:

Student Number: No.Marks

Instructions:

1. You have 50 minutes for this exam. You are not allowed to use any calculators.

2. This exam contains 3 questions and has a total of 100 marks.

1. (30 Marks)Evaluate each of the following indefinite integrals.

Solution. (a)

(b) 1. Factorizing the polynomial , we have

2. Get partial fraction decomposition.

Let .

Multipling the above equation by , we obtain

Taking x=1 in the above equation, we obtain A=1/2.

Taking x=0 in the above equation, we obtain C=-1/2.

Taking x=2 in the above equation, we obtain 5A+2B+C=1. This implies B=-1/2. Hence, we have

3. Evaluate the indefinite integral.

2. (35 Marks)Determine whether each of the following integrals converges or diverges. If it converges, find its value.

Solution. (a) .

(b) Let . Then x=1 is a singularity. Let . Then

We consider the following cases:

(i) When 1-p>0, and converges.

(ii) When 1-p<0, and diverges.

(iii) When p=1, . Hence, . In this case, diverges.

(35 Marks) Evaluate each of the following limits.

Solution. (a) .

(b)

Hence, .

Kunquan Lan
Thu Nov 9 10:46:22 EST 2000