Thomas Calculus 13th [Solutions]

Section 8.6 Integral Tables and Computer Algebra Systems 595
3.
x dx ( x 2) dx 1 1
2 dx x 2 dx 2 x 2 dx
x 2 x 2 x 2
3 1
x 2 x 2
2 2
2( x 2)
2 x 2 4
1 3 1 1 3
(We used FORMULA 11 with a 1, b 2, n 1 and a 1, b 2, n 1 )
C
x dx 1 (2x 3) dx 3 dx 1 dx 3 dx
(2x 3) 2 (2x 3) 2 (2x 3) 2 2x
3 2
2x
3
4.
3 2 3 2 3 2 3
1 1
1 3
2x
3 2x
3
1 2x 3 dx 3 2x 3 dx 1 2 3 2
C
2 2 2 2 1 2 2 ( 1)
1
( x 3)
(2x 3 3) C C
2 2x
3 2x
3
(We used FORMULA 11 with a 2, b 3, n 1 and a 2, b 3, n 3 )
5.
6.
3
3
3
1
x 2x 3 dx 1 (2x 3) 2x 3 dx 2x 3 dx 1 2x 3 dx 2x 3 dx
2 2 2 2
5 3
2x 3 2x 3 3 2 3 2
(2x 3) (2x 3) ( x 1)
1 2 3 2 C 2x
3 1 C C
2 2 5 2 2 3 2 5 5
(We used FORMULA 11 with a 2, b 3, n 3 and a 2, b 3, n 1)
3 2 3 2 5 3 2
5
5
3
x(7x 5) dx 1 (7x 5)(7x 5) dx (7x 5) dx 1 7x 5 dx 7x 5 dx
7 7 7 7
7 5
x x 5 2 5 2
x x x x
7 5 7 5
1 2 5 2
(7 5) 2(7 5) (7 5)
C 2 C 14 4 C
7 7 7 7 7 5 49 7 49 7
(We used FORMULA 11 with a 7, b 5, n 5 and a 7, b 5, n 3 )
9 4 ( 4)
7. x 9 4
dx
x dx C
2
x
x 2 x 9 4x
(We used FORMULA 14 with a 4, b 9 )
9 4x
1 9 4x
9
2 ln
x 9 9 4x
9
(We used FORMULA 13(b) with a 4, b 9 )
9 4x
2 9 4x
3
ln
x 3 9 4x
3
C
C
8. dx 4x
9 4 dx C
2
x 4x
9 ( 9) x 18 x 4x
9
(We used FORMULA 15 with a 4, b 9 )
4x
9 2 2 1 4 9
9 9 tan x C
x
9 9
(We used FORMULA 13(a) with a 4, b 9 )
4x
9 4 1
tan 4x
9 C
9x
27 9
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