8·3
9 9·1
6. 3 ¯ xe x dx 3e x ( x 1) C
1.
¯ cot x dx ln | sin x | C
2.
¯ (x 2)(3x 5) dx ln 3x 5 C
3.
¯ (ln x ) dx 2x 2x ln x x(ln x )
4.
¯ x cos x dx cos x x sin x C
5.
¯ ( x 2)
x 2
1
2
2
x
2
dx
C
2 ln | x 2 | C x 2
3 1 (10w 3) 2 C 15
7.
¯
8.
¯ t(t 5)
9.
¯x
10.
¯ sin u cos u du ln | tan u | C
10w 3 dw 1
dt t 5 ln | t 5 | C
x 2 dx
3 2(3x 4 ) ( x 2) 2 C 15
1
The definite integral 10
1.
¯
10
2.
¯
50
30
(3x 2 4 x 5) dx 1900 8 dx 640
x5 dx 596.25 2 x2 36 1 4. ¯ dt ln 6 6 t 3.
¯
7
¤5 ³ ¤5 ³ ¤ 5P ³ ³ 6 ¤ ¤ 5P ³ sec ¥ Q ´ tan ¥ Q ´ dQ ¥ sec ¥ ´ sec ¥ ´ ´ y 6.0221 5¦ ¦ 6 µ ¦6 µ ¦6 µ ¦ 12 µ µ 3 dx P 6. ¯ y 0.5236 1 2 6 4 x 2 103 7. ¯ (3x 4 5 x 3 21x 2 36 x 10) dx 5.15 1 20 5 ¤ 54 ln 5 54 ³ ¤ 34 ln 3 34 ³ 8. ¯ ( x 3 ln x ) dx ¥ ´ ¥ ´ y 195.2278 3 16 µ ¦ 4 16 µ ¦ 4 5.
¯
P
0.5P
¤ ³ ³ ¤P 1 P 1 cot 1 ( x )dx ¥ 3 ln(4 )´ ¥ ln(2)´ y 0.4681 6 2 µ µ ¦4 2 ¦ 5 1 10. ¯ dx 3 ln(1 e 5 ) ln(1 e 2 ) y 0.1202 2 1 ex 9.
9·2
¯
3
1
2
1. By Property 1, ¯ f ( x ) dx 0. 2
2. By Property 2, ¯
2 0
f ( x ) dx 12.
1
3. By Property 1, ¯ f ( x ) dx 0. 1
2
4. By Property 3, ¯ f ( x ) dx 27. 2 0
5. By Property 4, ¯ 5 f ( x ) dx 60. 2
9·3
1 d § x 2 (t 3) 5 dt ¶ 2 ¸· ( x 3)5 dx ©¨ ¯0 d § x 3t 5 dt ¶ 3x 5 2. ¸· dx ©¨ ¯1 1.
2
6. By Properties 4 and 3, ¯ 10 f ( x ) dx 270. 2
5
7. By Property 5, ¯ [ f ( x ) g ( x )]dx 14. 1
5
8. By Property 5, ¯ [ f ( x ) g ( x )]dx 30. 1
9. By Property 4, ¯
5
1
10. By Property 4,
¯
1 f ( x ) dx 4. 2
5
1
5
2 g ( x ) dx ¯ 3 f ( x ) dx 20. 1
d § x4 ¶ t sin t dt · 4 x 7 sin( x 4 ) dx ¨© ¯P ¸ 2 5 d § x 3 2 ¶ 4. t dt · 10 x 2 3 25 x dx ¨© ¯ 5 ¸ 3.
Answer key
113