MATH101 April 2011
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q1 (g) • Q1 (h) • Q1 (i) • Q1 (j) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q4 (a) • Q4 (b) • Q4 (c) • Q4 (d) • Q5 (a) • Q5 (b) • Q5 (c) • Q6 (a) • Q6 (b) • Q7 • Q8 •
Question 04 (a)
Evaluate the integral (hint: put ).
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.
If , then .
Use partial fractions.
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Set the following change of variable
then we have
Making this substitution in the integral gives
Now we use the method of partial fractions. We have
We want to find numbers and such that
Multiplying both sides by yields
Equating coefficients of powers of on both sides gives
From the first of these equations we see that . So the second equation becomes
Hence and . Therefore, we have
Using this to rewrite the last integral above, we find
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