Thomas Calculus 13th [Solutions]

Section 8.3 Trigonometric Integrals 573
38.
4 2 2 2 2 2 2 2
sec x tan x dx sec x tan x sec x dx tan x 1 tan xsec
x dx
4 2 2 2 1 5 1 3
tan xsec x dx tan xsec x dx tan x tan x C
5 3
39.
0 3
3 2 sec x dx;
2
u sec x, du sec x tan x dx, dv sec x dx, v tan x;
0 3 0 0 2 0
2
3 2 sec x dx 2sec x tan x
3
2 3 sec x tan x dx 2 1 0 2 2 3 2 3
sec x sec x 1 dx
0 3 0
4 3 2 sec x dx 2 sec x dx;
3 3
2 0 3 0 0 3
3 2 sec x dx 4 3 2ln sec x tan x
3
2 3
2 sec x dx 4 3 2ln |1 0 | 2ln 2 3
0 3
4 3 2ln 2 3 2 sec x dx 2 3 ln 2 3
40.
x 3 x x x x x 2 x x x
e sec e dx; u sec e , du sec e tan e e dx, dv sec e e dx, v tan e ;
x 3 x x x x 2 x x
e sec e dx sec e tan e sec e tan e e dx
x x x 2 x x
sec e tan e sec e sec e 1 e dx
x x 3 x x x x
sec e tan e sec e e dx sec e e dx
x 3 x x x x x
2 e sec e dx sec e tan e ln sec e tan e C
x 3 x
sec 1 x x x x
e e dx sec e tan e ln sec e tan e C
2
41.
4 2 2 2 2 2 1 3
sec d 1 tan sec d sec d tan sec d tan tan C
3
1 2 1 2
tan tan sec 1 C tan sec 2 tan C
3 3 3
42.
4 2 2 2 2 2
3sec (3 x) dx 1 tan (3 x) sec (3 x)3dx sec (3 x)3dx tan (3 x)sec (3 x)3dx
1 3
tan(3 x) tan (3 x)
C
3
43.
2 4 2 2 2 2 2 2 2 2
csc d 1 cot csc d csc d cot csc d cot cot
4 4 4 4 3
4
3 2
(0) 1
1 4
3 3
44.
6 4 2 2
2
2 4 2 2
sec x dx sec x sec x dx tan x 1 sec x dx tan x 2 tan x 1 sec x dx
4 2 2 2 2 1 5 2 3
tan x sec x dx 2 tan x sec x dx sec x dx tan x tan x tan x C
5 3
45.
3 2 2
4 tan x dx 4 sec x 1 tan x dx 4 sec x tan x dx 4 tan x dx 4 tan 2 x 4ln |sec x|
C
2
2 2 2 2 2
2 tan x 4ln |sec x| C 2 tan x 2ln sec x C 2 tan x 2ln 1 tan x C
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