Math 005B Final Exam Name:_____________________________________________________
Part I: Find the antiderivatives – 10 pts each:
1. ∫
cot(Ln(x))
3x
dx
2. ∫ sec(x) tan3
(x) dx
3. ∫ e
2xsin(3x) dx
4. ∫
x+2
x(x−4)
2
dx
5. ∫
Ln(x
3
)
x
2
dx
Part II: Evaluate the definite integrals (10 pts each)
6. ∫
x
3−2x
2
x
2+1
dx √2
0
7. ∫
1
√𝑥
2+4
dx √3
1
8. ∫ e
sin2(x)
cos(x)dx

3
π
3
– Remember you MUST show your work for any credit!
Part III Define the following – 2 pts each (You MUST use the definitions I
presented in class!):
9. Ln(x)
10. e
11. Arcsin(x)
12. sinh(x) and cosh(x)
Part IV: (8 pts total)
13. Put the Hyperbola in standard form, and find all of the features:
3×2 – 12x – y
2 – 8y + 4 = 0 Center:_______________
Vertices:___________ and _____________
Foci: ______________ and _____________
Asymptotes:____________________
and ___________________________
Part V: 8 pts each:
14. Find the limit: lim
x→0+
(cos(x) + 2x)
1
x
15. If 𝑟(𝜃) = 2 sin(3𝜗)find the equation of the tangent line to the curve at ϑ=
𝜋
4
Part VI (8 pts):
16. Derive the formula for the derivative of the Arccosecant function, using implicit
differentiation.
Part VII Points as indicated:
17. (15 pts) Recall that the Maclaurin series for f(x) = 1
1+𝑥
2
is ∑ (−1)
𝑛𝑥
∞ 2𝑛
𝑛=0
A) Find the radius and interval of convergence – be sure to check the end points!
B) Use the above fact to find the Maclaurin series for g(x) = Arctan(x)
18. Decide whether the series ∑
(−1)
nLn(n)
√n

n=2 diverges, converges absolutely, or
converges conditionally. Show all your steps and indicate the tests you are using!
(10 pts)
19. (10 pts) Find the first three non-zero terms in the Maclaurin series for
f(x) = cos(3x) (centered at x = 0, of course.)
20. (5 pts) Find the sum of the series: ∑
2
𝑛+1
3
3𝑛−2

𝑛=1