课程名称︰微积分甲上
课程性质︰土木系必修
课程教师︰张志中
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰102/12/3
考试时限（分钟）：30分钟
是否需发放奖励金：是
（如未明确表示，则不予发放）
试题 :
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