求微積分高手幫忙，微積分題目的解答和計算過程.

1.Limit(x->0)[√(5x+4) - 2]'/x'= Limit(x->0)[5/2√(5x+4)] / 1= 5/2√4= 5/4= (D)
2.Limit(x->0)(x^2-x-sin(x))'/(2x)'= [2x-1-cos(x)]/2= (-1-1)/2= -1= (B)
3.y = limit(x->oo)[(x^2-3) / (2x-4)]= limit(x->oo)[(x/2 + 1) + 1/(2x-4)]= x/2 + 1 + 1/oo= x/2 + 1 + 0=> y = x/2 + 1 = (C)
4. w'(3) = -1/2√(4-z)= -1/2√1= -1/2= (w-2)/(z-3)=> w = (z+7)/2 = (C)
5.y' = sec(x)*tan(x)= (sec(x))'*tan(x) + sec(x)*(tan(x))'= sec(x)*tan(x)^2 + sec(x)^3= (D)
6.y^2 + x^2 = y^4 - 2x2yy' + 2x = 4y^3*y' - 2y'(-2) = (x+1)/y(2y^2-1)= (-2+1)/1*(2*1-1)= -1= (B)
7. y(x) = x^3 - 3x^2 - 1y(3) = -1y^(-1) = x,y互換x = y^3 - 3y^2 - 11 = 3y^2*y' - 6yy'y'(3)= 1/3y(y-2)= 1/9*(3-2)= 1/9= (C)
8.y = x^(x+1)ln(y) = (x+1)*ln(x)y'/y = ln(x) + (x+1)*1/xy ' =y[ln(x) + (x+1)*1/x]= x^(x+1)*[ln(x) + (1+1/x)]= (B)
9.∫(0~π/2)sin(t)*cos(t)*dt= ∫sin(t)*d(sin(t))= (sin(t))^2 / 2= (sin90)^2 / 2 - (sin0)^2 / 2= 1/2= (C)
10.Σ(k=1~n)(n + 2k)= Σn + Σ2k= n^2 + 2n(n+1)/2= n^2 + n^2 + n= 2n^2 + n= (A)



2015-07-02 07:32:24 補充：
11.y = ∫(5~x)3t*sin(t)dt, dy(-pi/2)/dx = ?

dy/dx = d/dx{∫(5~x)3tsin(t)dt}

= 3x*sin(x) - d常數/dx

= -3*pi/2 * sin(-pi/2)

= 3*pi/2

= (C)


12.∫(0~4)√(2x+1)dx

= ∫√(2x+1)*d(2x+1)/2

= 1/2*∫√y*dy

= 1/2*2/3*y^1.5

= 1/3 * (2x+1)^1.5

= 1/3 * (27-1)

= 26/3

= (A)

2015-07-02 07:40:56 補充：
13.y = ∫(x+4)dx/(x^2+5x-6)

=∫(x+4)dx/(x+6)(x-1)

=∫(a/(x+6)+b/(x-1))dx

x+4 = a(x-1) + b(x+6)

=> a = 2/7, b = 5/7

=> y =∫2dx/7(x+6) + ∫5dx/7(x-1)

= 2/7 * ln|x+6| + 5/7 * ln|x-1|

= (A)

2015-07-02 07:42:36 補充：
14.∫(0~60)tan(x)*dx

= ∫sin(x)dx/cos(x)

= -∫d(cos(x))/cos(x)

= -ln(cos(x))

= -[ln(cos60) - ln(cos0)]

= -(ln(1/2) - ln(1))

= ln(2)

= (B)

2015-07-02 07:49:55 補充：
15.∫(0~180)sin(x/2)^3*dx

= -∫sin(x/2)^2*d(cos(x/2))

= ∫(cos(x/2)^2 - 1)d(cos(x/2))

= ∫(y^2 - 1)dy

= y^3/3 - y

= cos(x/2)^3 / 3 - cos(x/2)

= cos(90)^3/3 - cos(90) - cos(0)^3/3 + cos(0)

= 0 - 0 - 1/3 + 1

= 2/3

= (A)


16.∫(-1~1)|x-1|dx

= |x^2/2 - x|

= |1/2 * (1-1) - (1+1)|

= 2

= (D)

2015-07-02 08:02:30 補充：
超過2000字的限制 請另外增加題目

建議版主將問題切割成10個帖,每個帖2題,每個帖解答贈點5點,
這樣也許比較多人可以幫忙.

例如第一帖:
......pdf
請解第 1 , 2 題

也太多題,應該自己先寫看看 不會的題目在上來發問