ce amath --integration

Andrew

2007-10-02 5:27 am

1.please integrate the following:(1-cos2x)/(1+cos2x)w.r. x
2. suppose y=cot^3x-3cot x. Find dy/dx in terms of cot x. nHence find the integrate of cot^4x w.r.t x.

回答 (1)

雞尾包

2007-10-02 6:04 am

✔ 最佳答案

(1-cos2x)/ (1+cos2x)
= (2 sin^2 x) / (2 cos^2 x)
= tan^2 x
= sec^2 x - 1
int (sec^2 x - 1) dx
= tanx - x + C, C is constant
-----------------------------------------------------------------
y = cot^3 x - 3cot x
dy/dx
= (3cot^2 x - 3) d cot x/dx
= (3cot^2 x - 3)(-csc^2 x)
= -3(cot^2 x - 1)(1 + cot^2 x)
= -3(cot^4 x - 1)
dy/dx = -3cot^4 x + 3
y = -3 int cot^4 x dx + 3x + C, C is constant
cot^3 x - 3cot x = -3 int cot^4 x dx + 3x + C
3 int cot^4 x dx = -cot^3 x + 3cot x + 3x + C
int cot^4 x dx = -cot^3 x/3 + cot x + x + C/3
int cot^4 x dx = -cot^3 x/3 + cot x + x + c where c = C/3

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