Download 7.2

7.2: Trigonometric Integrals
Use identities and substitution to
evaluate trig integrals.
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Eval. sinm x cosn x dx: If n = 2k + 1
sinm x cos2k+1 x =
sinm x (1 − sin2 x)k cos x
So we
R use sub u = sin x, du = cos x dx.
Ex: cos3 x dx = sin x + 13 sin3 x + C .
If m = 2k + 1, switch roles of sin x and
cos x; use subRu = cos x, du = − sin x dx.
Ex: Evaluate sin3 x cos2 x dx
( 15 cos5 x − 13 cos3 x + C )
If m, n are both even use identities
cos2 x =
1 + cos 2x
,
2
sin2 x =
1 − cos 2x
2
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Show that cos2 x dx = x2 + sin42x + C
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and sin2 x cos2 x dx = x8 − sin324x + C
sin 2x = 2 sin x cos x may also be useful.
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Ex: sin3 x cos−2 x dx = sec x + cos x + C
Facts:
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2
R sec x dx = tan x + C ,
R sec x tan x dx = sec x + C ,
R tan x dx = ln | sec x| + C ,
sec x dx = ln | sec x + tan x| + C ,
sec2 x = tan2 x + 1.
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Eval. secm x tann x dx: If m = 2k > 0,
sec2k x tann x =
(tan2 x + 1)k−1 tann x sec2 x
So use sub u = tan x, du = sec2 x dx.
If n = 2k + 1 and m > 0
secm x tan2k+1 x =
secm−1 x (sec2 x − 1)k sec x tan x
So use sub u = sec x, du = sec x tan x dx.
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4
2
Ex: sec4 x tan x dx = tan4 x + tan2 x + C
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5
3
Ex: sec3 x tan3 x dx = sec5 x − sec3 x + C
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2
Ex: tan3 x dx = tan2 x − ln | sec x| + C