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8/14/2019 L16 101

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a x < y = f (x)

b[a ;[ y = f (x) [a ; b]

J (b) =b

a f (x)dx

limb

b

a

f (x)dx f (x)

[a ;[ lim

b

b

a

f (x)dx =

a

f (x)dx

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limbJ (b)

y = f (x) x = a

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y = f (x) ]; b] a]; b]

J (a ) =b

a

f (x)dx

limab

a

f (x)dx J (a )

lima

b

a

f (x)dx =b

f (x)dx.

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y = f (x) ] ;[ a] ;[

f (x)dx =

a

f (x)dx + a

f (x)dx

f (x)dx

8/14/2019 L16 101

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dx

1 + x2

dx1 + x 2

=

0

dx1 + x 2

+

0

dx1 + x 2

=

= lima

0

a

dx1 + x 2

+ limb

b

0

dx1 + x 2

= lima

arctgx|0a + limb

arctg x|b0 = .

[a ;[ f (x), (x) 0 f (x) (x)

a f (x)dx

a (x)dx

a (x)dx

a

f (x)dx

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1dx

x2

(1 + ex)

x 1 =1

x 2 (1 + ex) 1x 2

.

1

dxx 2 (1 + ex)

1

dxx 2

= 1x

1

= 1

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[a ;[ f (x), (x) 0 f (x) (x)

a f (x)dx

a (x)dx

1

x + 1x 3

x + 1x 3 xx 3 = 1x =

1

dxx = limb2

x b1 = 2 (b 1) = .

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a |f (x)

|dx

af (x)dx

a

f (x)dx

1sinxx 3 dx

sinxx 3

1x 3

=

1

dxx 3

= 1

2x 21

=12

=

1

sinxx 3

dx

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y = f (x) x[a ; c[ x = c

limbc0

b

a f (x)dx y = f (x) [a ; c[

c

a f (x)dx = limbc0b

a f (x)dx

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y = f (x) x

]a ; c] x = a

limba+0

c

b

f (x)dx y = f (x)

]a ; c] c

a

f (x)dx = limba+0

c

b

f (x)dx.

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y = f (x) [a ; b] x = xk , k = 1, n

b

a

f (x)dx =

x 1

a

f (x)dx +

x 2

x 1

f (x)dx + . . . +b

xn

f (x)dx

b

a

f (x)dx

b a

f (x)dx

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1

0dx

1 x. x = 1

1

0

dx1

x

= limb

1

0

b

0

dx1

x

= limb

1

0

21 xb0 = limb

1

02(1 b 1) = 2.

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1

1dx

x2 . x = 0

1

1

dxx 2

= lim1

0

1

1

dxx 2

+ lim2

+0

1

2

dxx 2

=

= lim101x

1

1+ lim2+0 1

1x

2

1= + .

1

1

dxx 2

= 1x

1

1

= 2.

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y = f (x), y = (x) [a ; c[ 0 f (x)(x) y = (x) y = f (x)

[a

;c[

y=

f (x

), y

=

(x

)

0 f

(x

) (x) y = f (x) y = (x)

[a ; c[ y = f (x)

c

a

|f (x)|dx

c

a f (x)dx c

af (x)dx

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c

adx

(x a )

f (x) =1

(x a ) x ]a ; c]

x = a

limb

a+0

c

b

(x a ) dx = limb

a+0

(x a ) +1

+ 1

c

b=

11

limb

a+0

1(x

a )1

c

b=

=1

1 1

(c a )1 1

1 lim

ba+01

(b a )1

=J = lim

ba+01

(b a )1=

1 < 0 . 1 > 0 = 1

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1

0dx

x + 4x 2

x = 0

f (x) =1

x + 4x 2