Evaluate the definite integrals

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 β€œexact” form, unless otherwise specified.

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

 

1. ∫(3π‘₯ βˆ’ 2)π‘’βˆ’π‘₯𝑑π‘₯

 

 

 

2. ∫ 𝐿𝑛(π‘₯)

π‘₯2 𝑑π‘₯

 

 

 

 

 

 

3. ∫√cos⁑(t)⁑sin3(t)dt

 

 

 

 

4. ∫tan4(x)⁑sec6(x)dx

 

 

 

 

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

5. ∫ π‘₯2βˆšπ‘Ž2 βˆ’β‘π‘₯2⁑𝑑π‘₯ π‘Ž

0

 

 

 

 

 

 

 

6. ∫ π‘₯3βˆ’4π‘₯+1

π‘₯2βˆ’3π‘₯+2 ⁑𝑑π‘₯

0

βˆ’1

 

 

 

 

 

 

 

7. ∫ 𝑒2π‘₯

1+𝑒π‘₯ ⁑𝑑π‘₯

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. ∫ π‘₯β‘π‘’βˆ’π‘₯ 2 ⁑𝑑π‘₯

∞

βˆ’βˆž

 

 

 

 

 

 

 

 

9. ∫ 1

π‘₯2βˆ’4 ⁑𝑑π‘₯

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

xβˆ’2 + etc.

π‘₯βˆ’3

π‘₯(π‘₯βˆ’2)3(π‘₯2+π‘₯+5)2

 

 

 

 

 

 

Part V:

11. (10 pts) Find the antiderivative: ∫ π‘₯+3

π‘₯2βˆ’4π‘₯+13 𝑑π‘₯

 

 

 

 

 

 

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

π‘₯βˆ’π‘’βˆ’π‘₯⁑ ⁑𝑑π‘₯

∞

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):

βˆ‘ 4π‘›βˆ’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) βˆ‘ 𝑛+7

(π‘›βˆ’2)2 ∞ 𝑛=3

 

 

 

 

B) βˆ‘ π‘›β‘π‘’βˆ’π‘›βˆžπ‘›=1

 

 

 

 

 

 

C) βˆ‘ βˆšπ‘›2+𝑛

𝑛4βˆ’5 ∞ 𝑛=3

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