IV. Summary

[ I.B ]

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sinn x dx∫ , cosn x dx∫

n even (n = 2p) --

write

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sinn x = sin2p x = (sin2 x)p = (1 − cos 2x2 )p = 12p(1− cos 2x)p , or

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cosn x = 12p(1+ cos 2x)p , expand algebraically and integrate term-by-term as

powers of cos 2x n odd (n = 2p+1) --

write

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sinn x = sinn−1 x ⋅ sin x = sin2p x ⋅ sin x = (1− cos2 x)p ⋅ sin x ,

use the substitution u = cos x , du = −sin x dx

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→ (1− u2)p ⋅ (−du) , and expand algebraically;

or

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cosn x = (1− sin2 x)p ⋅ cos x , u = sin x , du = cos x dx

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→ (1− u2)p ⋅ du any integer n > 1 -- can also use the reduction formulae

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sinn x dx = − 1n∫ sinn−1 x cos x + n−1n sinn−2 x dx∫ ,

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cosn x dx = 1n∫ cosn−1 x sin x + n−1

n cosn−2 x dx∫

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sinm x cosn x dx∫

n even ( n = 2q ) n odd ( n = 2q + 1 )

m even ( m = 2p ) I.D I.C.2 m odd ( m = 2p + 1 ) I.C.1 I.C.1 or I.C.2

I.C.1: write

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sinm x cosn x = sin2p x cosn x ⋅ sin x = (1− cos2 x)p cos2q x ⋅ sin x ,

use the substitution u = cos x , du = −sin x dx

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→ (1− u2)p ⋅ u2q ⋅ (−du) , and expand algebraically

I.C.2: write

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sinm x cosn x = sinm x cos2q x ⋅ cos x = sinm x (1− sin2 x)q ⋅ cos x ,

use the substitution u = sin x , du = cos x dx

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→ u2p ⋅ (1− u2)q ⋅ du , and expand algebraically

I.D: write

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sinm x cosn x = sin2p x cos2q x , which can be expressed as either

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(1− cos2 x)p ⋅ cos2q x or sin2p ⋅ (1− sin2 x)q ; expand algebraically and integrate

term-by-term as powers of sin2x or cos2x , using the results in I.B

special case ( m = n = 2p ) -- can also write

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sinm x cosm x = (sin x cos x)2p

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= (12 sin2x)2p = 1

22p(sin2 2x)p = 1

22p(1 − cos 4 x2 )p = 1

22p⋅12p(1− cos 4x)p

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= 123p

(1− cos 4x)p , expand algebraically and integrate term-by-term as

powers of cos 4x

[ II.B ]

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tann x dx∫ , secn x dx∫

(results are analogous for

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cot n x dx∫ , cscn x dx∫ )

secant (any integer n > 1) --

write

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secn x = secn−2 x ⋅ sec2 x and use integration by parts with u = secn−2 x ,

dv = sec2x dx ; the integration produces a term containing the original integral; can also be used to obtain the reduction formula

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secn x dx = 1n−1∫ secn−2 x tan x + n−2

n−1 secn−2 x dx∫

tangent (any integer n > 1) --

write

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tann x = tann−2 x ⋅ tan2 x = tann−2 x ⋅ (sec2 x −1) and use integration by

parts on one of the terms, with u = tann−2 x , dv = sec2x dx ; the integration produces a term containing the original integral; can also be used to obtain the reduction formula

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tann x dx = 1n−1∫ tann−1 x − tann−2 x dx∫

tangent for n even ( n = 2p ) --

can also write

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tann x = tan2p x = (sec2 x −1)p , expand algebraically and

integrate as powers of sec2 x , using the results above

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tanm x secn x dx∫ (results are analogous for

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cotm x cscn x dx∫ )

n even ( n = 2q ) n odd ( n = 2q + 1 )

m even ( m = 2p ) II.C II.E m odd ( m = 2p + 1 ) II.C or II.D II.D

II.C: write

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tanm x secn x = tanm x sec2q x = tanm x sec2q−2 x ⋅ sec2 x ,

use the substitution u = tan x , du = sec2x dx

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→ um ⋅ (u2 + 1)q−1 ⋅ du , and expand algebraically

II.D: write

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tanm x secn x = tan2p+1 x secn x = tan2p x secn−1 x ⋅ sec x tan x ,

use the substitution u = sec x , du = sec x tan x dx

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→ (u2 −1)2p ⋅ un−1 ⋅ du , and expand algebraically

II.E: write

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tanm x secn x = tan2p x secn x = (tan2 x)p secn x = (sec2 x −1)p secn x , expand algebraically and integrate term-by-term as powers of sec x , using the results in II.B

-- G. Ruffa 12-21 September 2003

revised and amended 25-26 January 2009