MATH 005B Exam #2, Chapter 7 and Chapter 11.1 – 11.4
Do as much as possible with no book or notes, as usual, then check your work.
Leave your answers in “exact” form, unless otherwise specified.
Part I: Find the antiderivatives (8 pts each):
1. ∫(3𝑥 − 2)𝑒−𝑥𝑑𝑥
2. ∫ 𝐿𝑛(𝑥)
Part II Evaluate the definite integrals (10 pts each):
5. ∫ 𝑥2√𝑎2 −𝑥2𝑑𝑥 𝑎
6. ∫ 𝑥3−4𝑥+1
7. ∫ 𝑒2𝑥
Part III: Solve these improper integrals by using the correct methods and steps –
show your steps – so you MUST use the definition! (10 pts each)
8. ∫ 𝑥𝑒−𝑥 2 𝑑𝑥
9. ∫ 1
10. (8 pts) Set up the partial fraction decomposition for the fraction below – you
do NOT need to solve for the constants – Leave it as A
x−2 + etc.
11. (10 pts) Find the antiderivative: ∫ 𝑥+3
12. (10 pts) Find the volume enclosed by the curve: f(x) = 1/x as x goes from 1 to
∞ if the curve is rotated about the line y = -2.
13. (10 pts) A) State the comparison test for integrals (limit at infinity version!)
B) Does the integral: ∫ 1
1 converge or diverge? Why? (You do NOT need
to find the value of the integral.)
14. If a0 = 3 and an+1 = (n+1)an, find a1, a2 and a3. (6 pts)
15. Find the exact value of the series, or explain why it diverges (8 pts):
16. Decide whether each of the following converges or diverges. State specifically
the test you are using, and demonstrate that test, carefully – in each case, you
must show the integral, or the series with which you are comparing, and why that
series or function satisfies the hypotheses of the test. (24 pts) – The only tests you
should need are Integral, comparison and limit comparison.
A) ∑ 𝑛+7
(𝑛−2)2 ∞ 𝑛=3
B) ∑ 𝑛𝑒−𝑛∞𝑛=1
C) ∑ √𝑛2+𝑛
𝑛4−5 ∞ 𝑛=3
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