評量5.36

1. 0.4772 + 0.4987 = 0.9759
3. + c
4. = = = ln2 + 3/2
5. = = 16/3
6. = =
7. = =
8. = = ln|y+4| + c
9. = = (1/5)x⁵ + 2x³ + 9x + c
10. = = x + ln|x| + c
11. = (1/2)ln|2x-1| + c
12. = =
13. 則: 則: c³ - c² - 6c + 6 = 0
    c = 1 or sqrt(6) or - sqrt(6)
    但 c = 1 or - sqrt(6)不合 ， ∴ c = sqrt(6)
14. = =

評量5.37

註:因為積分步驟因人而異，所以答案不是唯一，故此答案作為參考用。
將所算出之答案微分後，等於原積分之函數即可。

1. ∫(lnx/x)dx = ∫(lnx)d(lnx) = (1/2)(ln|x|)² + c
2. ∫(e^sinxcosx)dx = ∫(e^sinx)d(sinx) = e^sinx + c
3. ∫(cos²xsin⁵x)dx = -∫(cos²xsin⁴x)d(cosx) =
   -∫[cos²x(1-cos²x)²]d(cosx) = ∫[-cos²x + 2cos⁴x - cos⁶x]d(cosx) =
   -(1/3)cos³x + (2/5)cos⁵x - (1/7)cos⁷x + c
4. ∫[(x+7)(3-2x)^(1/3)]dx = (-1/2)∫[y^(1/3)(17/2 - (1/2)y)]dy (令y = 3-2x)=
   (3/28)(3-2x)^(7/3) - (51/16)(3-2x)^(4/3) + c
5. ∫[3x/(x² - 2x - 8)]dx + ∫(3/2)d(ln|x² - 2x - 8| + ∫[3/(x+2)(x-4)]dx =
   ∫[3x/(x² - 2x - 8)]dx + ∫[(-1/2)/(x+2)]dx + ∫[(1/2)/(x-4)]dx =
   (3/2)ln|x² - 2x - 8| - (1/2)ln|x+2| + (1/2)ln|x-4| + c
   另解:2ln|x+4| + ln|x-2| + c
6. ∫[1/(x² - 6x + 12)]dx = ∫{1/[(x-3)² + 3]}dx =
   ∫[√3/(3y² + 3)]dy (令x = 3 + √3y)= (√3/3)tan^-1[(x-3)/√3] + c
7. = (令 y = lnx => x = e^y)= =
   = = 3(e² - e)
8. = = 1
9. (1/7)(5⁷ - 3⁷)
10. 如圖設a,b分別為下圖中的半長軸和半短軸長，則橢圓面積 = 4 X 藍色部分面積
    由橢圓一般式 : x²/a² + y²/b² = 1 知 :
    y = b(1 - x²/a²)^(1/2)
    則藍色部分面積 = (令x = asinT)= =
    
    所以橢圓面積 =