課程名稱︰微積分甲(上)
課程性質︰系定必修
課程教師︰黃漢水
開課學院：生機、生工、地質、工管
開課系所︰數學系
考試日期（年月日）︰200/1/14
考試時限（分鐘）：170分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
一 Find the following intergral: (30％)
0 1 ______
(1)∫{[5x^3-4x^2+7x]/[(x-1)^2*(x^2+1)]}dx (2)∫(8t/√2t^2+1)dt
-1 0
0
(3)∫[t^2*(1+t)^9]dt
-1
_________
二 Find the length of curve y=√(x^2+2)^3/3 from x=0 to x=3 (15％)
三 Let D be the region bound by the curves y=㏑x , x=1 and y=1 (15％)
(1)Find the centroid of the region D
(2)Find the volume of the solid generated by revolving the region D
about the line x=1
e
四 Let In(x)=∫[2x(㏑x)^n]dx, Jn=∫[2x(㏑x)^n]dx for any n=1,2,3... (20％)
1
(1)Find the relations between In(x),In-1(x)
(2)Find the relations between Jn,Jn-1
(3)Find the values J1,J2,J3,J4
五 (1)Find a real number k such that
_____ _____ _____
∫(√x^2-9/x^2)dx=㏑│x+√x^2-9│+k√x^2-9/x+c (10％)
5 _____
(2)Find the integral ∫(√x^2-9/x^2)dx (5％)
3
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