Operacion Quantitativa De Negocio

C H A P T E R Integration
Section 4.1 Section 4.2 Section 4.3 Section 4.4 Section 4.5 Section 4.6

4

Antiderivatives and Indefinite Integration . . . . . . . . . 177 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Riemann Sums and Definite Integrals . . . . . . . . . . . 188 The Fundamental Theorem of Calculus . . . . . . . . . . 192 Integration by Substitution . . . . .. . . . . . . . . . . . 197 Numerical Integration . . . . . . . . . . . . . . . . . . . 204

Review Exercises

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

C H A P T E R Integration
Section 4.1

4

Antiderivatives and Indefinite Integration

Solutions to Odd-Numbered Exercises

1.d 3 dx x3 dy dt y

C

d 3x dx

3

C

9x

4

9 x4

3.

d 1 3 x dx 3

4x

C

x2

4

x

2 x

2

5.

3t2 t3 C C 3t2

7.

dy dx y

x3

2

d 3 t Check: dt

2 5 x 5

2

C
2

Check:

d 2 5 x dx 5

C

x3

2

Given 9.
3

Rewrite x1 3 dx

Integrate x4 3 4 3 x
1 2

Simplify 3 4 x 4
3

x dx

C

C

11.

1 x x

dx

x

3 2dx

1 2 1 x 2 2 2

C

2 x 1 4x2

C

13.

1 dx 2x3 x2 2 3x

1 x 2

3

dx

C

C

15.

x

3 dx d x2 dx 2

3x C

C x 3

17.

2x Check:

3x 2 dx d 2 x dx

x2 x3

x3 C

C 2x 3x 2

Check:

19.

x3 Check:

2 dx d 1 4 x dx 4

1 4 x 4 2x

2x C

C x3 2

21.

x3

2

2x d 2 5 x dx 5

1 dx
2

2 5 x 5 x2 x

2

x2 C

x x3
2

C 2x1

Check:

23.

3

x2 dx

x2 3 dx d 3 5 x dx 5
3

x5 3 5 3 C x2
3

C
3

3 5 x 5 x2

3

C

25.

1 dx x3 Check: d dx

x

3

dx

x

2

2 C

C 1 x3

1 2x2

C

Check:

1 2x2

177

178

Chapter 4

Integration

27.

x2

x x

1

dx
2

x3 2 3 x 3

2

x1

2

x
2

1 2

dx x3
2

2 5 x 5 x1

2

2 3 x 3 x

2

2x1 x22

C x x 1

2 1 x 15

2

3x2

5x

15

C

Check:

d 2 5 x dx 5

2

2x1

C

2

1 2

29.

x

1 3x

2 dx x3

3x2 1 2 x 2 C

x

2 dx 2x 3x2 x C x 1 3x 2 2

31.

y2 y dy Check: d 2 7 y dy 7

y5 2 dy
2

2 7 y 7 y5

2

C y2 y

C

2

Check:

d 3 x dx

1 2 x 2

2x

33.

dx Check:

1 dx d x dx

x C

C 1

35.

2 sin x Check: ddx

3 cos x dx 2 cos x

2 cos x 3 sin x C

3 sin x 2 sin x

C 3 cos x

37.

1 Check:

csc t cot t dt d t dt csc t

t C

csc t 1

C csc t cot t

39.

sec2 Check:

sin d tan d

d cos

tan C

cos sec2

C sin

41.

tan2 y

1 dy C

sec2 y dy sec2 y

tan y tan2 y

C 1

43. f x
y

cos x

d Check: tan y dy

3 2

C
2C

3
x
2

2

0 2

3C

45. f x f x

2 2x
y
5 4

47. f x C
f )x)
2x 2

1 x
x3 3

x2 x3 3
x
y
4

49. C

dy dx y 1

2x 2x 1 x2
2

1, 1, 1 1 dx 1 x 1 x2 C ⇒ C x C 1

f x
f )x)

f )x)

x3 3

x

3

y

f′

3

f )x)

2x

3 2

x
3 2 1 2 3 3 2 1 3 2

x

Answers will vary.

f

Answers will vary.

Section 4.1

Antiderivatives and Indefinite Integration

17951.

dy dx y 4 y

cos x, 0, 4 cos x dx sin 0 sin x sin x C 4

53. (a) Answers will vary.
y 5

C ⇒ C 4

−3

x

5

−3

(b)

dy dx y 2 2 y

1 x 2 x2 4 42 4 C x2 4 8t3 8t3 4 2 5t
3 2

1, 4, 2 x 4 C
−4

6

8 −2

C

x

2

55. f x f x f0 f x

4x, f 0 4x dx 6 2x 2 20 6
2

6 2x 2 C 6

57. h t ht h1 ht 61. f x f 4 f 0

5, h 1 5 dt 5 11 2t4

4 5t C 11

C⇒CC⇒C

2t4 x 2 0 x 2 2

59. f x f 2 f 2 f x f 2 f x f x f 2 f x

2 5 10 2 dx 4 2x 2x 6 x
2

2x

C1 1

f x f 4 x 4 C2 f x f x f 0 f x 0

3 2

dx C1 3
1 2

2x

1 2

C1 3

2 x

C1

C1 1

5 ⇒ C1

2 ⇒ C1

1 dx

x2

2 x 2x 0 4x1
2

C2 x

10 ⇒ C2 4

3 dx

4x1

2

3x

C2

C2 3x

0 ⇒ C2 4 x

0 3x

63. (a) h t h0 ht (b) h 6 0

1.5t 0

5 dt...