Calc 1 Worksheet #26
Integrating Even and Odd powers of Sine and Cosine
______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Notes:
For odd powers: PEEL one of the sines or cosines, then substitute using:
sin2x = 1 – cos2x and cos2x = 1 – sin2x.
For even powers: Use the substitutions:
1
and
sin2 = 1 − cos 2
2
(
)
cos2
=
(
1
1 + cos 2
2
)
_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
1. ∫ cos 2 (2 x)dx
2. ∫ sin 3 (3x)dx
3. ∫ cos 4 xdx
______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
x
4. ∫ sin 2 dx
2
5. ∫ sec 2 (4 x)dx
6. ∫ sin 2 x cos xdx
______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
π
2
7. ∫ 5sin (2 x)dx
2
8. ∫ cos x sin xdx
3
9.
∫ cos xdx
3
0
______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
π
2
10.
∫ sin xdx
2
0
11. ∫ tan 2 ( 4x ) dx
12. If f '( x) = 3x 2 − 2 and if f(2) = 14,
Hint: use Trig Identity
then f(1) = ?
______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
sin( x + h) − sin( x)
=?
h→0
h
13. lim
⎛ esin 3 x ⎞
dy
14. Find f '( x) if f ( x) = ln ⎜
if y = tan−1 (3x)
⎟ 15. Find
dx
⎝ cos x ⎠
Hint: Simplify first
______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Answers:
1
1
x + sin 4 x + C
2
8
1
1
4. x − sin x + C
2
2
5
5
7. x − sin 4 x + C
2
8
1.
10.
π
4
13. cos(x)
Revised: 10/18/2010
1
1
2. − cos3 x + cos3 3 x + C
3
9
1
5. tan 4 x + C
4
−1 4
8.
cos x + C
4
1
11. tan(4 x) − x + C
4
14. 3cos(3x) + tan x
3
sin 2 x sin 4 x
x+
+
+C
8
4
32
1
6. sin 3 x + C
3
2
9.
3
12. f(1) = 9
3.
15.
3
1 + 9x 2