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.

3. Show all of your work. Your work must justify the answer that you give.

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 5*A*+2*B*+*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, .

Thu Nov 9 10:46:22 EST 2000