Calculus (I)

W^EN-C^HING L^IEN

Department of Mathematics National Cheng Kung University

2008

Case 1: Unbounded Intervals

Let f(x)be continuous on the intervals[a,∞).

If

zlim→∞

Z z

a

f(x)dx exists and has a finite value.

we say that the improper integral Z ^∞

a

f(x)dx

converge and define

Z ^∞ Z z

Let f(x)be continuous on the intervals[a,∞).

If

zlim→∞

Z z

a

f(x)dx exists and has a finite value.

we say that the improper integral Z ^∞

a

f(x)dx

converge and define

Z ^∞ Z z

Let f(x)be continuous on the intervals[a,∞).

If

zlim→∞

Z z

a

f(x)dx exists and has a finite value.

we say that the improper integral Z ^∞

a

f(x)dx

converge and define

Z ^∞ Z z

Let f(x)be continuous on the intervals[a,∞).

If

zlim→∞

Z z

a

f(x)dx exists and has a finite value.

we say that the improper integral Z ^∞

a

f(x)dx

converge and define

Z ^∞ Z z

Let f(x)be continuous on the intervals[a,∞).

If

zlim→∞

Z z

a

f(x)dx exists and has a finite value.

we say that the improper integral Z ^∞

a

f(x)dx

converge and define

Z ^∞ Z z

If f (x) is continuous on (a, b] and lim

_x→a⁺

f (x) does not exist, we define

Z

b

a

f (x )dx = lim

c→a⁺

Z

c

b

f (x )dx

If f (x) is continuous on (a, b] and lim

_x→a⁺

f (x) does not exist, we define

Z

b

a

f (x )dx = lim

c→a⁺

Z

c

b

f (x )dx

If f (x) is continuous on (a, b] and lim

_x→a⁺

f (x) does not exist, we define

Z

b

a

f (x )dx = lim

c→a⁺

Z

c

b

f (x )dx

If f ( x ) is continuous on [ a, b ) and lim

_x→b−

f ( x ) does not exist, we define

Z

b

a

f (x)dx = lim

c→b⁻

Z

c

a

f (x)dx

If f ( x ) is continuous on [ a, b ) and lim

_x→b−

f ( x ) does not exist, we define

Z

b

a

f (x)dx = lim

c→b⁻

Z

c

a

f (x)dx

If f ( x ) is continuous on [ a, b ) and lim

_x→b−

f ( x ) does not exist, we define

Z

b

a

f (x)dx = lim

c→b⁻

Z

c

a

f (x)dx

Case 1:

1

Z

_∞

1

1 x dx ,

Z

_∞

1

1 x

²

dx ,

Z

_∞

1

1 x

^p

dx

2

Z

_∞

−∞

e

^−|x|

dx

3

Z

_∞

e

1

x (ln x )

²

dx

∞

Case 1:

1

Z

_∞

1

1 x dx ,

Z

_∞

1

1 x

²

dx ,

Z

_∞

1

1 x

^p

dx

2

Z

_∞

−∞

e

^−|x|

dx

3

Z

_∞

e

1

x (ln x )

²

dx

∞

Case 1:

1

Z

_∞

1

1 x dx ,

Z

_∞

1

1 x

²

dx ,

Z

_∞

1

1 x

^p

dx

2

Z

_∞

−∞

e

^−|x|

dx

3

Z

_∞

e

1

x (ln x )

²

dx

∞

Case 1:

1

Z

_∞

1

1 x dx ,

Z

_∞

1

1 x

²

dx ,

Z

_∞

1

1 x

^p

dx

2

Z

_∞

−∞

e

^−|x|

dx

3

Z

_∞

e

1

x (ln x )

²

dx

∞

Case 1:

1

Z

_∞

1

1 x dx ,

Z

_∞

1

1 x

²

dx ,

Z

_∞

1

1 x

^p

dx

2

Z

_∞

−∞

e

^−|x|

dx

3

Z

_∞

e

1

x (ln x )

²

dx

∞

Case 1:

1

Z

_∞

1

1 x dx ,

Z

_∞

1

1 x

²

dx ,

Z

_∞

1

1 x

^p

dx

2

Z

_∞

−∞

e

^−|x|

dx

3

Z

_∞

e

1

x (ln x )

²

dx

∞

1

Z

2

0

1

(x − 1)

¹³

dx

2

Z

e

1

1 x ln x dx

3

Z

1

0

√ 1

x dx

Z

1

1

1

Z

2

0

1

(x − 1)

¹³

dx

2

Z

e

1

1 x ln x dx

3

Z

1

0

√ 1

x dx

Z

1

1

1

Z

2

0

1

(x − 1)

¹³

dx

2

Z

e

1

1 x ln x dx

3

Z

1

0

√ 1

x dx

Z

1

1

1

Z

2

0

1

(x − 1)

¹³

dx

2

Z

e

1

1 x ln x dx

3

Z

1

0

√ 1

x dx

Z

1

1

(1)

Assume that f(x)≥0 for x ≥a.

Suppose that there exists g(x)such that g(x)≥f(x)for x ≥a andR^∞

a g(x)dx is convergent. Then Z ^∞

a

f(x)dx is also convergent

(1)

Assume that f(x)≥0 for x ≥a.

Suppose that there exists g(x)such that g(x)≥f(x)for x ≥a andR^∞

a g(x)dx is convergent. Then Z ^∞

a

f(x)dx is also convergent

(1)

Assume that f(x)≥0 for x ≥a.

Suppose that there exists g(x)such that g(x)≥f(x)for x ≥a andR^∞

a g(x)dx is convergent. Then Z ^∞

a

f(x)dx is also convergent

(1)

Assume that f(x)≥0 for x ≥a.

Suppose that there exists g(x)such that g(x)≥f(x)for x ≥a andR^∞

a g(x)dx is convergent. Then Z ^∞

a

f(x)dx is also convergent

(2)

Assume that f(x)≥0 for x ≥a.

Suppose that there exists g(x)such that g(x)≤f(x)for x ≥a andR^∞

a g(x)dx is divergent. Then Z ^∞

a

f(x)dx is also divergent

(2)

Assume that f(x)≥0 for x ≥a.

Suppose that there exists g(x)such that g(x)≤f(x)for x ≥a andR^∞

a g(x)dx is divergent. Then Z ^∞

a

f(x)dx is also divergent

(2)

Assume that f(x)≥0 for x ≥a.

Suppose that there exists g(x)such that g(x)≤f(x)for x ≥a andR^∞

a g(x)dx is divergent. Then Z ^∞

a

f(x)dx is also divergent

(2)

Assume that f(x)≥0 for x ≥a.

Suppose that there exists g(x)such that g(x)≤f(x)for x ≥a andR^∞

a g(x)dx is divergent. Then Z ^∞

a

f(x)dx is also divergent

1

Z ^∞

1

√ 1

1+x⁶dx

2

Z ^∞

−∞

1 e^x +e^−x

3

Z ^∞

1

1 px+√

xdx

4

Z ^∞ 1

√ dx

1

Z ^∞

1

√ 1

1+x⁶dx

2

Z ^∞

−∞

1 e^x +e^−x

3

Z ^∞

1

1 px+√

xdx

4

Z ^∞ 1

√ dx

1

Z ^∞

1

√ 1

1+x⁶dx

2

Z ^∞

−∞

1 e^x +e^−x

3

Z ^∞

1

1 px+√

xdx

4

Z ^∞ 1

√ dx

1

Z ^∞

1

√ 1

1+x⁶dx

2

Z ^∞

−∞

1 e^x +e^−x

3

Z ^∞

1

1 px+√

xdx

4

Z ^∞ 1

√ dx

1

Z ^∞

1

√ 1

1+x⁶dx

2

Z ^∞

−∞

1 e^x +e^−x

3

Z ^∞

1

1 px+√

xdx

4

Z ^∞ 1

√ dx