Thomas Calculus 13th [Solutions]

Section 8.5 Integration of Rational Functions by Partial Fractions 585
56.
1
2
1 1 1
1 2
V x tan x dx x tan x dx u tan x, du 1 dx, dv x dx,
v 1 x
0 0
2
1 x
2
1 1 2
1 1
1 2 1 1
x tan x 1 x dx 1 tan 1 0 1 1 1 dx 1 1 1 dx
2 2 0
2 2 2 0
2 8 2 0
2
0 1 x 1 x 1 x
1 1 1
1 1 1 1 1 1 1 1 1
( 2)
dx dx x tan x
tan 1 0 0
0 0 1 x
0
8 2 2 2
8 2 2 8 2 2 4
2 2 2
3 2
57. (a) Integration by parts: u x , du 2 x dx, dv x 1 x dx, v 1 1 x
3
(b)
3 2 2 2
3 2
2
3 2
2 2
3 2
2
5 2
x 1 x dx 1 x 1 x 1 1 x 2x dx 1 x 1 x 2 1 x C
3 3 3 15
2 2
Substitution: u 1 x x 1 u du 2x dx 1 du x dx
2
3 2 2 2 1 1 3/2 1 3/2 1 5/2
x 1 x dx x 1 x x dx 1 u u du u u du u u C
2 2 3 5
2
3/2
2
5/2
1 1 x 1 1 x C
3 5
(c) Trig substitution:
2
x sin , , dx cos d , 1 x cos
2 2
3 2 3 2 2 2 2
x 1 x dx sin cos cos d sin cos sin d 1 cos cos sin d
2 4 3 5 2
3 2
2
5 2
cos sin d cos sin d 1 cos 1 cos C 1 1 x 1 1 x C
3 5 3 5
58. (a) The slope of the line tangent to y f ( x ) is given by f ( x ). Consider the triangle whose hypotenuse is
(b)
the 30 ft rope, the length of the base is x and height
2
900 x
x
f ( x)
,
thus
900 x
f ( x) .
x
2
900 x
x
dx
2
2
h 900 x . The slope of the tangent line is also
2
x 30sin , 0 , dx 30cos d , 900 x 30cos
2
2 1 sin
30cos 30cos d 30 cos d 30 d 30 csc d 30 sin d
30sin sin sin
30 900
2
30ln csc cot 30cos C x
30ln 900 ;
x x
x C
2
2
2 2
f (30) 0 0 30ln 30 900 30 2 30 900
2
30 30
900 30 C C f ( x) 30ln x
x x
900 x
8.5 INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
1. 5x 13 A B 5x 13 A( x 2) B( x 3) ( A B) x (2A 3 B)
( x 3)( x 2) x 3 x 2
A B 5
B (10 13) B 3 A 2; thus, 5x
13 2 3
2A
3B
13
( x 3)( x 2) x 3 x 2
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