math exam

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)

𝑥2 𝑑𝑥

3. ∫√cos⁡(t)⁡sin3(t)dt

4. ∫tan4(x)⁡sec6(x)dx

Part II Evaluate the definite integrals (10 pts each):

5. ∫ 𝑥2√𝑎2 −⁡𝑥2⁡𝑑𝑥 𝑎

0

6. ∫ 𝑥3−4𝑥+1

𝑥2−3𝑥+2 ⁡𝑑𝑥

0

−1

7. ∫ 𝑒2𝑥

1+𝑒𝑥 ⁡𝑑𝑥

1

0

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. ∫ 𝑥⁡𝑒−𝑥 2 ⁡𝑑𝑥

∞

−∞

9. ∫ 1

𝑥2−4 ⁡𝑑𝑥

3

2

Part IV:

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 +

B

x−2 + etc.

𝑥−3

𝑥(𝑥−2)3(𝑥2+𝑥+5)2

Part V:

11. (10 pts) Find the antiderivative: ∫ 𝑥+3

𝑥2−4𝑥+13 𝑑𝑥

Part VI:

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

𝑥−𝑒−𝑥⁡ ⁡𝑑𝑥

∞

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):

∑ 4𝑛−2

52𝑛+1

∞

𝑛=1

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) ∑ 𝑛⁡𝑒−𝑛∞𝑛=1

C) ∑ √𝑛2+𝑛

𝑛4−5 ∞ 𝑛=3