課程名稱︰微積分甲上
課程性質︰土木系必修
課程教師︰張志中
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰102/12/3
考試時限（分鐘）：30分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Calculas A2 (102 Fall) Quiz Five
17:50~18:20, December 3, 2013
It's necessary to explain all the reasons in detail and show all of your
work on the answer sheet. Or you will NOT get any credits. If you used any
theroems in textbook or proved in class, state it carefully and explicitly.
1. (20 pts) Find the values of p for which the integral
∞ 1
∫ ———— dx
e p
x(lnx)
converges and evaluate the integral for those values of p.
2. (40 pts) Determine whether the following integrals are convergent of
divergent. Explain why.
∞ x
(a) ∫ ———— dx
0 3
x + 1
-1 6
∞ (tan x)
(b) ∫ ————— dx
0 x
2+e
3. (40 pts) Evaluate the following integrals.
10
(a) ∫—————— dx
3 2
x -x +9x-9
2
x +1
(b) ∫——————— dx
2 2
(x -2x +2)
Answer:
1. p>1, 1/(p-1)
2. (a)(b) Both are convergent.
3.
(a)
1 2 1 -1 x
ln|x-1|-—ln(x +9)-—tan — + C
2 3 3
(b)
1 -1 x-3
—(3tan (x-1)+————) + C
2 2
x -2x+2