# Evaluate the definite integrals

MATH 005B Exam #2, Chapter 7 and Chapter 11.1 β 11.4

Do as much as possible with no book or notes, as usual, then check your work.

Leave your answers in βexactβ form, unless otherwise specified.

Part I: Find the antiderivatives (8 pts each):

1. β«(3π₯ β 2)πβπ₯ππ₯

2. β« πΏπ(π₯)

π₯2 ππ₯

3. β«βcosβ‘(t)β‘sin3(t)dt

4. β«tan4(x)β‘sec6(x)dx

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

5. β« π₯2βπ2 ββ‘π₯2β‘ππ₯ π

0

6. β« π₯3β4π₯+1

π₯2β3π₯+2 β‘ππ₯

0

β1

7. β« π2π₯

1+ππ₯ β‘ππ₯

1

0

Part III: Solve these improper integrals by using the correct methods and steps β

show your steps β so you MUST use the definition! (10 pts each)

8. β« π₯β‘πβπ₯ 2 β‘ππ₯

β

ββ

9. β« 1

π₯2β4 β‘ππ₯

3

2

Part IV:

10. (8 pts) Set up the partial fraction decomposition for the fraction below β you

do NOT need to solve for the constants β Leave it as A

x +

B

xβ2 + etc.

π₯β3

π₯(π₯β2)3(π₯2+π₯+5)2

Part V:

11. (10 pts) Find the antiderivative: β« π₯+3

π₯2β4π₯+13 ππ₯

Part VI:

12. (10 pts) Find the volume enclosed by the curve: f(x) = 1/x as x goes from 1 to

β 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: β« 1

π₯βπβπ₯β‘ β‘ππ₯

β

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):

β 4πβ2

52π+1

β

π=1

16. Decide whether each of the following converges or diverges. State specifically

the test you are using, and demonstrate that test, carefully β 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) β The only tests you

should need are Integral, comparison and limit comparison.

A) β π+7

(πβ2)2 β π=3

B) β πβ‘πβπβπ=1

C) β βπ2+π

π4β5 β π=3

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