236
CHAPTER 28
28.39
28.40
28.41
28.42
28.43
28.44
28.45
28.46
28.47
28.48
28.49
Find a reduction formula for J sec" x dx for n a 2.
Use the reduction formula of Problem 28.39 to find J sec
3 x dx.
Use the reduction formula of Problem 28.39 to find J sec
4 x dx.
Establish a reduction formula for / x"e" dx for n a 1.
Apply the reduction formula of Problem 28.42 and the result of Problem 28.20 to find J x
3 e
3 * dx.
Find a reduction formula for J x" sin ax dx for n > 1.
Use the reduction formula of Problem 28.44 and the solution of Problem 19.33 to find J x
2 sin x dx.
Prove the reduction formula
Let M = x, dv = VI + x dx, du = dx,
Find J xVl +x dx.
Let M = x,
Apply the reduction formula of Problem 28.47 to find
Then
Find J cos x"
3 d*, using Problem 28.6.
First make the substitution
x = w
3
,
dx = 3w
2 dw.
Then
Jcosjc"
3 dx = 3 J w
2 cos wdw =
3[(w2 - 2) sin w + 2w cos w] + C = 3(x2'3 - 2) sin xln + 6*"3 cos *"3 + C.
Let M = sec"
2 x, dv = sec
2 A; d*, du = (n- 2) sec"
3 A: sec * tan x dx, v= tan *. Then / sec" x dx =
tanjcsec""2jc-(n-2) J sec""2 * tan2 *<& = tan jsec"'2 x - (n -2) J sec""2 x(sec2 x - 1) dx
tan x sec"~
2 x - (n - 2) J sec" x dx + (n - 2) / sec""
2 x dx. Solving for J sec" x dx,
(tan x sec x + In |sec x + tan *|) + C.
sec*
Let w = x", dv = e°* dx, du = nx"~
l dx, v = (1 /a)e". So
Let M = ;C", l dx, v =—(I/a) cos ax. Then J *" sin ax
cos a* dje.
cos ax +
J jr
2 sin x dx = —x
2 cos x + 2 / je cos * dx = —or
2 cos x + 2(x sin x + cos *) + C = 2x sin x + cos x (2 — x
2 ) + C.