Download 8.2 Trigonometric Integrals 443 These identities come from the

8.2
Trigonometric Integrals 443
These identities come from the angle sum formulas for the sine and cosine functions (Section 1.3). They give
functions whose antiderivatives are easily found.
EXAMPLE 8 Evaluate
sin 3x cos 5x dx.
Solution
From Equation (4) with m = 3 and n = 5, we get
sin 3x cos 5x dx =
~ j [sin ( — 2x) + sin 8x] dx
=
I2
(sin 8x
+ C.
Powers of Sines and Cosines Evaluate the
integrals in Exercises 1-22.
—
sin 2x) <ix
cos 8x . cos 2x
16
vr + sinxdx
cosx
27. / —^^d.
V3
VT
sin2 20 cos3 20
22. /
3
cos3 20 sin 20 dd
cos 2x dx cos x sin x dx
sin3 x dx sin5 x dx cos3 x
dx 11.
47.
29. /
J5TT/
/6
J sin x cos x dx
3
3
IT
1 — sinx 1
— sinx'
/
cot3 x dx
: dx
vT
JTT/6
/2
flVT- cos 20 dO
31. /
J tan x dx49.
(1 -
32. /
cos203/2
13. ^ cos2 x dx
15. /
sin ydy Jo
17. /
8 sin4 x dx Jo
, / 16 sin2 x cos2 x dx 8
—
3 sin
dx
Evaluate the integrals in Exercises 33-50.
4. / sin4 2x cos 2x dx 6. /
M/if: Multiply by
37r/4
cos3 4x dx
Vl - sin2xdx
30. /
JTT/2
sin —
8. /
/o
dx
2
/-TT/6
3 cos5 3x dx Jo
10. /
.J
sec2 x tan2 x dx
d/9
25. /
Vl - sin2 tdt
J
34.
J sec x tan x
dx 36.
11 / e r f 2*am***
23. / J^^dx
38.
2
y
sec3 x
CTT/2
16.
y
sin2 x dx
tan3 x dx 38. / sec4 x tan2
7 cos1 tdt 18. / 8
x dx
40. / <?x sec <?x dx
33. / sec2 x tan x dx 35.
Evaluate the integrals in Exercises 23-32
42. / 3 sec 3x dx
277
/ sec x tan • • 37, 39,
3
cos4 27TX dx
sec3 x dx
r/3
8 sin >'cos ydy Jo
20. /
CTT/2
!
y sec 0d0
4
-./*
f ir j2
Jo
I;
41.
24. /
26,
VT — cos2xdx Jo
44. / sec x dx
/71/4
46. /
6 tan4 x dx
7-7T/4
43. / esc4 0 dO 45. /
48. / cot6 2x dx 50.
4 tan3 x <£r
/ 8cot4/dr