数学Paper 1讨论专区

lonely_world

confuse when got trigo come out. who can teach me again?

Log

回复 1361# lonely_world
∫sin x(1+cos²x)dx = ∫ sin x +sinx cos²x dx
                          = ∫ sin x dx  - ∫ -sin x (cos x)² dx
                          = - cos x - (cos x)³/ 3 + c
                          =  - cos x - (cos³ x )/3 +c

海涵

该怎么做？？教教我。。
If the normal at P(ap^2  ,  2ap) to the parabola y^2 = 4ax meets the curve again at Q(aq^2 , 2aq) , prove that p^2 + pq +2 = 0.
Prove that the equation of locus of the point of intersection of the tangent to the parabola at P and Q is [y^2(x+2x)] + 4a^3 = 0.

海涵

这题怎么做？？。。。

P(p^2 , 2p) , Q(q^2 , 2q) are two points on the curve y^2 = 4x.
Prove that the equation of the chord joining P and Q is (p+q)y -2x = 2pq.
Deduce the equation of the tangent to the curve at P.
The chord PQ passes through the fixed point (3,0) , prove that the midpoint of the chord lies on the curve y^2 = 2(x-3). The tangent to the curve at P and Q meet at T, show that the locus of T is a straight line parallel to the y-axis.

Razor_1130

dx
∫------------
   2 + x^(1/2)


x^(1/2)= square root x                    let u^2 = x
那位大大可以教导小弟下嘛。。头痛

Log

回复 1365# Razor_1130

let  u²=x---> u=x½
x=u²   
dx/du=2u --->dx= 2u du
∫ dx/(2+x½ )= ∫ (2u du) /(2+u)
                 =2 ∫ u/(2+u) du
                 = 2  ∫ 1 -  2/(u+2)  du  --------》long division
                  = 2[u - 2 ln lu+2l] +c
                 = 2u -4 ln lu+2l +c
∴                = 2x½ - 4 ln l x½+2 l +c

Allmaths

该怎么做？？教教我。。
If the normal at P(ap^2  ,  2ap) to the parabola y^2 = 4ax meets the curve a ...
海涵 发表于 5-7-2010 02:54 PM

[y^2(x+2x)] + 4a^3 = 0      <-----这个应该有错误





注:有错请纠正.谢谢!

peaceboy

[y^2(x+2x)] + 4a^3 = 0
Allmaths 发表于 5-7-2010 06:48 PM

    好厉害

peaceboy

该怎么做？？教教我。。
If the normal at P(ap^2  ,  2ap) to the parabola y^2 = 4ax meets the curve a ...
海涵 发表于 5-7-2010 02:54 PM

P(p^2 , 2p) , Q(q^2 , 2q) are two points on the curve y^2 = 4x.
Prove that the equation of the chord joining P and Q is (p+q)y -2x = 2pq.
Deduce the equation of the tangent to the curve at P.
The chord PQ passes through the fixed point (3,0) , prove that themidpoint of the chord lies on the curve y^2 = 2(x-3). The tangent tothe curve at P and Q meet at T, show that the locus of T is a straightline parallel to the y-axis.

m = 2q-2p / q^2-p^2
   = 2(p-q) / (p+q)(p-q)
   = 2/(p+q)
y= 2x / (p+q) +c
2p -2p^2/(p+q) = c
c = (2p^2+2pq - 2p^2) / (p+q )   
  = 2pq/(p+q)
y = 2x+2pq / p+q
(p+q) y = 2x+2pq
(p+q)y -2x = 2pq.  proven

==================================

y^2 =4x
2y dy/dx = 4
dy/dx = 2/y
         = 2/2p
         =1/p
y-2p = 1/p ( x-p^2)
y=x/p+p equation of tangent

=====================================



(p+q)y -2x = 2pq. from proven
when x=3 , y=0
2pq = -6 ----@

mid pnt. = [ (p^2 + q^2)/2 , (2p+2q)/2 ]
   x=(p^2 + q^2)/2
   y= p+q

x/y = (p^2 + q^2)/2(p+q)
2(p+q)x = (p^2+q^2) y
             =[ (p+q)^2 -2pq ] y
             = [(p+q)^2 + 6] y
sub p+q =y
2xy =[(y)^2 + 6] y
y^2 = 2x-6
       = 2(x-3)

==============================================
y=x/p+p
y=x/q+q
x/p+p =x/q+q
x(q-p)/pq = q-p
x=pq
from @
2pq = - 6
pq = -3
x= -3
y= p+q
since x=-3 for all value of y
therefore that is a straight line parallel to the y-axis.

Allmaths

这题怎么做？？。。。

P(p^2 , 2p) , Q(q^2 , 2q) are two points on the curve y^2 = 4x.
Prove that  ...
海涵 发表于 5-7-2010 03:18 PM

Allmaths

P(p^2 , 2p) , Q(q^2 , 2q) are two points on the curve y^2 = 4x.
Prove that the equation of the  ...
peaceboy 发表于 5-7-2010 08:54 PM

彼此都厉害...

Razor_1130

10 dx
∫ ----------------
     25+x^2

谢谢log，还有一个问题。。
再次真心的谢谢你

芭樂

回复 1371# Allmaths


    需要互夸吗？？

peaceboy

10 dx
∫ ----------------
     25+x^2

谢谢log，还有一个问题。。
再次真心的谢谢你
Razor_1130 发表于 6-7-2010 04:31 PM

    很难下，好像要用trigo

http://www.numberempire.com/integralcalculator.php

回复  Allmaths


    需要互夸吗？？
芭樂 发表于 6-7-2010 05:30 PM

骗下积分可以换鲍鱼

Allmaths

10 dx
∫ ----------------
     25+x^2

谢谢log，还有一个问题。。
再次真心的谢谢你
Razor_1130 发表于 6-7-2010 04:31 PM

Razor_1130

Allmaths 发表于 6-7-2010 07:41 PM

通了！！谢谢。。
就是sub进去的那刻我断了。。
谢谢

Razor_1130

还有一题

integral 2^x dx

谢谢

Allmaths

还有一题

integral 2^x dx

谢谢
Razor_1130 发表于 6-7-2010 08:53 PM

这样的integration学校应该很少教...

Allmaths

还有一题

integral 2^x dx

谢谢
Razor_1130 发表于 6-7-2010 08:53 PM

或者

Razor_1130

或者
Allmaths 发表于 6-7-2010 09:53 PM

    这个方法我明白多了！谢谢。。
不懂你方便再为我做多一题吗？

integral sin^-1 (x/3) dx