Thomas Calculus 13th [Solutions]

554 Chapter 8 Techniques of Integration
45. Since secant is an even function and the domain is symmetric to 0, x 0.
For the y-coordinate:
y
1
2
/4 2 1
/4
sec x dx
/4 2
tan x
/4
/4 /4
sec x dx ln sec x tan x
/4
/4
1
ln 2 1 ln 2 1
ln
1 1
2 1 ln 3 2 2
2 1
0.567
46. Since both cosecant and the domain are symmetric around / 2, x / 2.
y
1 5 /6 2 1
5 /6
csc x dx
/6 2
cot x
2
/6
5 /6 5 /6
csc x dx ln csc x cot x
/6
/6
1 3 3
2
ln 2 3 ln 2 3
3 3
0.658
2 3 ln 7 4 3
ln
2 3
47.
48.
3 3
3 x x
1 3x e dx xe C
1
dx
2
1 sin x
Multiply the integrand by
sec
sec
2
2
x
.
x
2 2
1 sec x
sec x
dx dx dx
2 2 2 2
1 sin x sec x tan x 1 2 tan x
u tan x,
du sec x dx
2
sec x
1
dx
du
2 2
1 2 tan x 1 2u
2
v 2 u, dv 2 du
1 1 1
du
1 2u
2 1 v
1 tan
1
v
2
2 2
1 tan
1 2 tan
2
dv
C
x
C
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