Giai Bai Tap Tich Phan Suy Rong p2

GII BI TP TCH PHN SUY RNG

Bi ging in t

TS. L Xun i

Trng i hc Bch Khoa TP HCM

Khoa Khoa hc ng dng, b mn Ton ng dng

Email: [email protected]

TP. HCM 2015.

TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 1 / 36

Tch phn suy rng loi 1 Tnh tch phn suy rng loi 1

Tnh tch phn suy rng loi 1

Cu 1

Tnh tch phn suy rng I =

+1

dx

xx2 + x + 1

.

t t =1

x x = 1

t dx = dt

t2.

i cn

x 1 +t 1 0

.



Khi

I =

01

dtt2

1t .

1t2+ 1t + 1

=

10

dt1 + t + t2

=

10

d(t + 12

)(t + 12

)2+ 34

=

[ln

t + 12 +t2 + t + 1]1

0

= ln

(3

2+3

) ln

(3

2

)= ln

(1 +

23

).




Cu 2

Tnh tch phn suy rng I =

+1

arctan x

x2dx .

t

u = arctan xdv = dxx2

du =

dx

1 + x2

v = 1x



Khi

I =

[1x. arctan x

]+1

+

+1

dx

x(1 + x2)=

=pi

4+

+1

dx

x

+1

xdx

1 + x2=pi

4+

[ln |x | 1

2ln(1 + x2)

]+1

=

=pi

4+

[ln

x1 + x2]+

1

=pi

4+ ln 1 ln 1

2=pi

4+

ln 2

2


Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t

ngha hnh hc

Trong trng hp f (x) > 0,x [a,+), gi trca tch phn suy rng hi t c ngha hnh hc

l din tch ca hnh phng v hn c gi hn

bi x = a, trc Ox v th hm f (x)



Ch .

T ngha hnh hc ca tch phn suy rng, ta

c nu tn ti gii hn hu hn v khc 0

limx+ f (x) = A 6= 0

v f (x) kh tch trn mi on [a, b] [a,+)th tch phn suy rng

+a

f (x)dx phn k



1

Nu > 1 th I =+a

dx

xhi t.

2

Nu 6 1 th I =+a

dx

xphn k.

1

Nu > 1 th I =+2

dx

x. ln xhi t.

2

Nu < 1 th I =+2

dx

x. ln xphn k.

3

Nu = 1 th I =+2

dx

x. ln xhi t nu

> 1, phn k nu 6 1.



1

Nu > 1 th I =+a

dx

xhi t.

2

Nu 6 1 th I =+a

dx

xphn k.

1

Nu > 1 th I =+2

dx

x. ln xhi t.

2

Nu < 1 th I =+2

dx

x. ln xphn k.

3

Nu = 1 th I =+2

dx

x. ln xhi t nu

> 1, phn k nu 6 1.TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 8 / 36


Cu 3

Tm tch phn sau hi t I =

+1

dx

x 31 + x2

1

x 31 + x2

x+ 1x.x2/3

=1

x+2/3

tch phn I hi t th +2

3> 1 > 1

3.



Cu 4

Tm tch phn sau hi t

I =

+1

(2x + 3)dx

(4 + x) 31 + x4

Trng hp 1: > 0

(2x + 3)

(4 + x) 31 + x4

x+ 2xx.x4/3

=2

x+1/3

tch phn I hi t th +1

3> 1 > 2

3.



Trng hp 2: = 0

(2x + 3)

(4 + x) 31 + x4

x+ 2x5.x4/3

=2

5x1/3

I phn k v1

3< 1

Trng hp 3: < 0

(2x + 3)

(4 + x) 31 + x4

x+ 2x4.x4/3

=1

2x1/3

I phn k v1

3< 1.

Vy tch phn I hi t th > 2/3.



Trng hp 2: = 0

(2x + 3)

(4 + x) 31 + x4

x+ 2x5.x4/3

=2

5x1/3

I phn k v1

3< 1

Trng hp 3: < 0

(2x + 3)

(4 + x) 31 + x4

x+ 2x4.x4/3

=1

2x1/3

I phn k v1

3< 1.

Vy tch phn I hi t th > 2/3.TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 11 / 36


Cu 5

Tm tch phn sau hi t

I =

+1

(3x + 4x)dx(5 + x)1

Trng hp 1: > 0

(3x + 4x)(5 + x)1

x+ 4xx(1)

=4

x(1)1

tch phn I hi t th

( 1) 1 > 1 > 2 < 1 > 2.



Cu 5

Tm tch phn sau hi t

I =

+1

(3x + 4x)dx(5 + x)1

Trng hp 1: > 0

(3x + 4x)(5 + x)1

x+ 4xx(1)

=4

x(1)1

tch phn I hi t th

( 1) 1 > 1 > 2 < 1 > 2.TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 12 / 36


Trng hp 2: = 0

(3x + 4x)(5 + x)1

x+ 4x61

=4

61.x1

I phn k v 1 < 1

Trng hp 3: < 0

(3x + 4x)(5 + x)1

x+ 4x51

=4

51.x1

I phn k v 1 < 1Vy tch phn I hi t th > 2.



Trng hp 2: = 0

(3x + 4x)(5 + x)1

x+ 4x61

=4

61.x1

I phn k v 1 < 1Trng hp 3: < 0

(3x + 4x)(5 + x)1

x+ 4x51

=4

51.x1

I phn k v 1 < 1

Vy tch phn I hi t th > 2.



Trng hp 2: = 0

(3x + 4x)(5 + x)1

x+ 4x61

=4

61.x1

I phn k v 1 < 1Trng hp 3: < 0

(3x + 4x)(5 + x)1

x+ 4x51

=4

51.x1

I phn k v 1 < 1Vy tch phn I hi t th > 2.



Cu 6

Tm tch phn sau hi t

I =

+3

ex + ln x(1 + x)2

dx

Trng hp 1: > 0

ex + ln x(1 + x)2

x+ ln xx(2)

=1

x(2). ln1 x

tch phn I hi t th ( 2) > 1 > 1 +2 < 12 > 1 +2.



Cu 6

Tm tch phn sau hi t

I =

+3

ex + ln x(1 + x)2

dx

Trng hp 1: > 0

ex + ln x(1 + x)2

x+ ln xx(2)

=1

x(2). ln1 x

tch phn I hi t th ( 2) > 1 > 1 +2 < 12 > 1 +2.TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 14 / 36


Trng hp 2: = 0

ex + ln x(1 + x)2

x+ ln x22

=1

22.x0 ln1 x

I phn k v 0 < 1

Trng hp 3: < 0

ex + ln x(1 + x)2

x+ ln x1

=1

x0 ln1 x

I phn k v 0 < 1

Vy tch phn I hi t th > 1 +2.



Cu 7

Tm tch phn sau hi t

I =

+2

exdx

(x 1) ln x

Trng hp 1: > 0

ex

(x 1) ln xx+ e

x

x ln xx+ +

Do , tch phn I phn k.



Trng hp 2: = 0

ex

(x 1) ln xx+ 1

ln x=

1

x0. ln1 x

I phn k v 0 < 1

Trng hp 3: < 0

ex

(x 1) ln x =(x 1)ex

ln xx+ x

ex

ln x

xexln x1x2

=x+2

ex . ln xx+ 0

Vy tch phn I hi t th < 0.



Trng hp 2: = 0

ex

(x 1) ln xx+ 1

ln x=

1

x0. ln1 x

I phn k v 0 < 1

Trng hp 3: < 0

ex

(x 1) ln x =(x 1)ex

ln xx+ x

ex

ln x

xexln x1x2

=x+2

ex . ln xx+ 0

Vy tch phn I hi t th < 0.TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 17 / 36


ngha hnh hc

Trong trng hp f (x) > 0,x [a, b), gi trca tch phn suy rng hi t c ngha hnh hc


bi x = a, x = b trc Ox v th hm f (x),

trong x = b l tim cn ng ca hm s f (x)



ngha hnh hc

Trong trng hp f (x) > 0,x (a, b], gi trca tch phn suy rng hi t c ngha hnh hc


bi x = a, x = b trc Ox v th hm f (x),

trong x = a l tim cn ng ca hm s f (x)


Tch phn suy rng loi 2 nh ngha tch phn

ba f (x)dx, c [a, b] l im gin on

Nu hm s f (x) khng b chn khi x c, vic (a, b) th tch phn suy rng b

a

f (x)dx =

ca

f (x)dx +

bc

f (x)dx




nh l

Tch phn suy rng

ba

f (x)dx c gi l hi t

nu c 2 tch phn

ca

f (x)dx v

bc

f (x)dx u hi

t khng ph thuc ln nhau.




1

Nu < 1 th tch phnba

dx

(b x) vba

dx

(x a) hi t

2

Nu > 1 th tch phnba

dx

(b x) vba

dx

(x a) phn k




Cu 8


I =

31

dx4x x2 3

I =

31

d(x 2)1 (x 2)2 = arcsin(x 2)

31=

= arcsin 1 arcsin(1) = pi2+pi

2= pi.




Cu 9


I =

20

dx

(x 1)x2 x + 1

I =

10

dx

(x 1)x2 x + 1+2

1

dx

(x 1)x2 x + 1 = I1+I2

Tch phn I hi t khi c hai tch phn I1, I2 cng hi t.




1

(x 1)x2 x + 1x1 1

(x 1)12 1 + 1

=1

(x 1)1Tch phn I1 phn k. Vy I cng phn k.



Cu 10

Tm tch phn sau hi t I =

pi0

1 cos xx

dx .

1 cos xx

x0+x2

2

x=

1

2x2

tch phn I hi t th 2 < 1 < 3.



Cu 11

Tm tch phn sau hi t

I =

10

e2 + x2 ecos x

xdx .

e2 + x2 ecos x

xx0+

e[(

1 + 12 .x2

e2

)(1 x22

)]x

=1 + e2

2e.

1

x2

tch phn I hi t th 2 < 1 < 3.



Cu 11

Tm tch phn sau hi t

I =

10

e2 + x2 ecos x

xdx .

e2 + x2 ecos x

xx0+

e[(

1 + 12 .x2

e2

)(1 x22

)]x

=1 + e2

2e.

1

x2

tch phn I hi t th 2 < 1 < 3.TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 27 / 36


Cu 12

Tm tch phn sau hi t

I =

20

xdx3

(x 1)(x 2)2 .



Trng hp 1: > 0

I =

10

xdx3

(x 1)(x 2)2+3/21

xdx3

(x 1)(x 2)2+

+

23/2

xdx3

(x 1)(x 2)2 .



x

3

(x 1)(x 2)2x1+ 1

(x 1)1/3x

3

(x 1)(x 2)2x1 1

(x 1)1/3x

3

(x 1)(x 2)2x2 2

(x 2)2/3Do , tch phn I hi t.



Trng hp 2: = 0

I =

10

dx3

(x 1)(x 2)2+3/21

dx3

(x 1)(x 2)2+

+

23/2

dx3

(x 1)(x 2)2 .



13

(x 1)(x 2)2x1+ 1

(x 1)1/31

3

(x 1)(x 2)2x1 1

(x 1)1/31

3

(x 1)(x 2)2x2 1

(x 2)2/3Do , tch phn I hi t.



Trng hp 3: < 0

I =

1/20

dx

x 3(x 1)(x 2)2 +

11/2

dx

x 3(x 1)(x 2)2+

+

3/21

dx

x 3(x 1)(x 2)2 +

23/2

dx

x 3(x 1)(x 2)2 .



1

x 3

(1 x)(x 2)2x0+ 1

34.x

1

x 3

(x 1)(x 2)2x1+ 1

(x 1)1/31

x 3

(x 1)(x 2)2x1 1

(x 1)1/31

x 3

(x 1)(x 2)2x2 2

(x 2)2/3Do , tch phn I hi t khi < 1 > 1.TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 34 / 36


Vy tch phn I hi t th{ > 0

1 < < 0 > 1.



THANK YOU FOR ATTENTION


Tch phn suy rng loi 1Tnh tch phn suy rng loi 1Tm tch phn suy rng loi 1 hi t

Tch phn suy rng loi 2Tnh tch phn suy rng loi 2nh ngha tch phn ab f(x)dx, c [a,b] l im gin onTm tch phn suy rng loi 2 hi t

Documents

Giai Bai Tap Tich Phan Suy Rong p2