Indefinite Integral

2008-12-01 5:36 am

1a.Show that d/dx cot^(n-1) x = -(n-1)cot^n x -(n-1)cot^(n-2) x
b. Hence show that S(cot^n x)dx = -S[cot^(n-2) x] -cot^(n-1) x / (n-1)

2a. Let y= cos x sin^(n-1) x, where n is a positive interger. Find dy/dx.
Hence show that
S(sin^n x)dx = -cos x sin^(n-1) x +(n-1)S[sin^(n-2) xos^2 x]dx...........(*)
b.By writing cos^2 x = 1-sin^2 x in (*), show that
S(sin^n x)dx = -cosx sin^(n-1) x /n + (n-1)/n

回答 (1)

Yin Fai

2008-12-01 7:46 am

✔ 最佳答案

1a. cot x = cosx/sinx
d(cotx)/dx
= [sinx(-sinx)-cosx(cosx)]/sin2x
= -1-cot2x ..... (#)

d(cotn-1x)/dx
= (n-1) cotn-2x [d(cotx)/dx] <<<<< Chain Rule
= (n-1) cotn-2x [ -1-cot2x ] <<<<< By (#)
= -(n-1) cotn-2x -(n-1) cotnx
= RHS

1b. Take Integral in both sides on part a.
S d(cotn-1x) = S [-(n-1) cotn-2x -(n-1) cotnx] dx
S d(cotn-1x) = S [-(n-1) cotn-2x]dx - S [(n-1) cotnx] dx
(n-1) S(cotnx) dx = -(n-1) S(cotn-2x)dx - cotn-1x
.....

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