Calculus II Final Exam, May 4, 2005, Exams of Calculus

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MA 126: Calculus II Final Test; May 4, 2005

Time limit: 150 min.

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Each problem is worth of 10 points.

1. Evaluate the indefinite integral

∫ sin 1 x x^2

dx.

2. Evaluate the definite integral

∫ (^9)

1

t ln tdt.

4. Evaluate the definite integral

∫ (^2)

0

x − 1

x^2 + 3x + 2

dx.

5. Evaluate the improper integral

∫ (^) ∞

0

x 2 e −x^3 dx.

7. Determine if the series

∑^ ∞

n=

n^2 + 3n + 2

is convergent or divergent. If convergent, find its sum.

8. Determine if the series

∑^ ∞

n=

n^4 − n^3

is convergent or divergent.

10. Find the Maclaurin series representation of the function

f (x) =

1 − x^3

1 + x^3

and determine the interval of convergence.

11. Find the radius of convergence and the interval of convergence for the power

series

∑^ ∞

n=

xn

n^33 n^

13. Find the Taylor series expansion of f (x) = ln x centered at a = 3. (You don’t

have to show that Rn(x) → 0.)

14. Evaluate the indefinite integral

∫

e −x^2 dx

as a power series.