integration presentation

8/8/2019 Integration Presentation

1/40

I n t e g r a t i o n - I n d e n i t e

C o n c e p t s a n d P r o b l e m s

V i d y a l a n k a r I n s t i t u t e

L e c t u r e s l i d e s

M a t h e m a t i c s @ V i d y a l a n k a r I n s t i t u t e I n t e g r a t i o n - I n d e n i t e
http://find/http://goback/


2/40

O u t l i n e



3/40

I n t e g r a t i o n a s r e v e r s e o f D i e r e n t i a t i o n

D e n i t i o n

A f u n c t i o n F ( x ) i s c a l l e d t h e a n t i d e r i v a t i v e a o f t h e f u n c t i o n f ( x ) o nt h e i n t e r v a l

[a,

b]

i f a t a l l p o i n t s o f t h e i n t e r v a l F

(x

) =f

(x

).

a

N o t e i f t h e a n t i d e r i v a t i v e e x i s t s t h e n i t s n o t u n i q u e , t h e r e i s a f a m i l y o f

a n t i d e r i v a t i v e f o r a f u n c t i o n w h i c h d i e r s a m o n g b y a c o n s t a n t

D e n i t i o n

I f F(

x)

i s t h e a n t i d e r i v a t i v e o f f(

x)

t h e n F(

x) +

c i s d e n e d a s t h e

i n d e n i t e i n t e g r a l o f f (x ) d e n o t e d a s f ( x ) d x , t h u s w e h a v e f ( x ) d x = F ( x ) + C i f F ( x ) = f ( x )



4/40

E x i s t e n s e o f a n t i d e r i v a t i v e s

E x a m p l e

I s t h i s f u n c t i o n d e r i v a t i v e o f a n y f u n c t i o n c o n t i n u o u s o n [1 , 1 ]

f(

x) =

0 ,1 x < 01 , 0 x 1

T h e o r e m

C o n t i n u i t y o f f o n [

a,

b]

i s s u c i e n t f o r f t o h a v e a n a n t i d e r i v a t i v e

o n[

a,

b]

b u t n o t e c o n t i n u i t y i s n o t t h e n e c e s s a r y

a

c o n d i t i o n .

a

D e r i v a t i v e o f i s n o t c o n t i n u o u s , w h e r e ( x ) =

x

2

s i n

1

x

, x = 00 , x = 0

D e r i v a t i v e o f ( x ) e x i s t s ( s a y (x )) a t a l l p o i n t s b u t i s n o t c o n t i n u o u s o n R ,m e a n s (x ) i s n o t c o n t i n u o u s b u t i s d e r i v a t i v e o f ( x )



5/40

P r o p e r t i e s o f D e n i t e i n t e g r a t i o n

T h e o r e m

1

i f F ( x ) = f ( x ) t h e n f ( x ) d x

= ( F (x ) + C ) = F ( x ) = f ( x )

2

D i e r e n t i a l o f i n t e g r a l i s t h e i n t e g r a n d

d

f

(x

)d x

=

f(

x)

d x

3

I n d e n i t e i n t e g r a l o f d i e r e n t i a l o f s o m e f u n c t i o n i s e q u a l t o

t h i s f u n c t i o n p l u s a r b i t r a r y c o n s t a n t d F ( x ) = F ( x ) + C



6/40

B a s i c i n t e g r a l s f r o m d e r i v a t i v e s

B a s i c I n t e g r a l s

1

d

d x

x

n = n x n 1 x n d x = x n +1n + 1

+ c

2

d

d x

s i n x = c o s x c o s x d x = s i n x + c3

d

d x

c o s x

= s i n x s i n x d x = c o s x + c4

d

d x

t a n x=

s e c

2

x

s e c

2

x d x=

t a n x+

c

5

d

d x

c o t x =

c s c

2

x

c s c

2

x d x =

c o t x + c

6

d

d x

s e c x = s e c x t a n x s e c x t a n x d x = s e c x + c7

d

d x

c s c x=

c s c x c o t x

c s c x c o t x d x

= c s c x

+c



7/40

B a s i c i n t e g r a l s f r o m d e r i v a t i v e s

B a s i c I n t e g r a l s

1

d

d x

e

x = e x e x d x = e x + c2

d

d x

a

x = a x l n a

a

x

d x =a

x

l n a

+ c

3

d

d x

l n | x | = 1x

1x

d x = l n |x |+ c

4

d

d x

s i n

1

x =1

1

x

2

11

x

2

d x = s i n 1 x + c

5

d

d x

t a n 1

x

=

1

1 + x 2 1

1 + x 2d x

=t a n

1

x

+c

6

d

d x

s e c

1

x=

1

| x | x 2 1

1

|x | x 2 1 d x = s e c

1

x+

c



8/40

P r o b l e m s - e l e m e n t a r y i n t e g r a l s

E x a m p l e s

1 (

1+

x) 3

x xd x

2

s e c

2

x c o s e c

2

x d x

3

x

2 + c o s 2 x

1 + x 2c o s e c

2

x d x

4

( s e c x + t a n x )2

d x

5

a

m x

b

n x

d x

6

t a n x

s e c x + t a n xd x

7

s i n 2 x s i n 3 x d x



9/40

P r o p e r t i e s o f i n d e n i t e i n t e g r a l s

T h e o r e m

1

I n t e g r a l o f s u m o f t w o o r m o r e f u n c t i o n s i s s u m o f t h e i r

i n t e g r a l s

f

(x

) g ( x ) d x = f (x ) d x g (x ) d x2

I n t e g r a l o f s c a l a r m u l t i p l e o f a f u n c t i o n i s s c a l a r m u l t i p l e o f t h e

i n t e g r a l o f t h a t f u n c t i o n

c f ( x ) d x = c f (x ) d x



10/40

M e t h o d s o f i n t e g r a t i o n - S u b s t i t u t i o n

S u b s t i t u t i o n

I f w e n e e d t o e v a l u a t e f (x ) d x

w e c a n s u b s t i t u t e x= (t ) = d x = ( t ) d t

f ( x ) d x = f (( t ))( t ) d tC o r o l l a r y

1

I f F

( x ) = f ( x ) = f (a x + b ) d x =1

a

F ( a x + b ) + c

U s i n g t h i s e x t e n d a l l t h e e l e m e n t a r y f o r m u l a e

2

f

( x )f

(x

)d x

=l n|

f(

x)|+

c

3 ( f ( x )) n f ( x ) d x = ( f ( x ))n +1

n

+1

+ c

1M a t h e m a t i c s @ V i d y a l a n k a r I n s t i t u t e I n t e g r a t i o n - I n d e n i t e


11/40

P r o b l e m s - S u b s t i t u t i o n

E x a m p l e s

1

x + 3

( x + 2 ) 2d x

2

8 x + 1 3

4 x+

7

d x

3

I f

1

x+

x

5

d x=

f(

x) +

c t h e n e v a l u a t e

x

4

x+

x

5

d x



13/40

E x t e n d e d f r o m e l e m e n t a r y

E x t e n d e d f r o m e l e m e n t a r y t h a t c a n b e f u r t h e r e x t e n d e d

1

1

a

2

x

2

d x=

s i n

1

x

a

+c p u t x

=a s i n

2

1

x

2 +a

2

d x = l n x +x 2 + a 2 + c p u t x = a t a n

3

1

x

2 a 2 d x = l n

x +

x

2 a 2

+ c p u t x = a s e c



14/40

I n t e g r a t i o n b y p a r t s

T h e o r e m

F o r u a n d v f u n c t i o n s o f x , t h e d i e r e n t i a l o f t h e i r p r o d u c t i s

d (u v ) = u d v v d u =

u d v = u v

v d u

E x t e n d e d f r o m e l e m e n t a r y a n d u s e d f u r t h e r t o e x t e n d

1

a

2 x 2 d x = x2

a

2 x 2 + a2

2

s i n

1

x

a

+ c u s i n g I B P

2

x

2 + a 2 d x =x

2

x

2 + a 2 +a

2

2

l n x +x 2 + a 2 + c I B P3

x

2 a

2

d x=

x

2

x

2 a

2 a2

2

l n

x +x 2 a 2 + c I B P4

e

x (f

(x

) +f

(x

))d x

=e

x

f(

x) +

c



15/40

P r o b l e m s - I n t e g r a t i o n b y p a r t s

E x a m p l e s

1

x e

x

d x

2

x3

s i n x d x

3

s i n

1

x d x

4

e

x

s i n x d x

5

x l n x d x

6

es i n

1

x

d x

7

e

a x

s i n b x d x

8

( f ( x ) g ( x ) f ( x ) g ( x )) d x

9

P r o v e t h e f o r m u l a e

g e n e r a t e d i n t h e p r e v i o u s

s l i d e

1 0 e x x( x + 1 )

2

d x



16/40

B o o k l e t P r o b l e m s - S u b s t i t u t i o n & I n t e g r a t i o n b y p a r t s

E x a m p l e s

1 + c o s 4 xc o t

x t a n xd x

s i n

3

xc o s

x

2

d x

1x (x

n

+ 1 )d x

1

(a 2 x 2 )3 /2 d x

F o r a n y n a t u r a l n u m b e r m , e v a l u a t e

x

3 m +x

2 m +x

m

2 x

2 m +3 x

m +6

1

/m

d x

w h e r e x >

0

I f 4 e x+

6e

x

9e

x 4

e

x

d x = A x + B l n (9 e2 x 4 ) + C

t h e n n d t h e v a l u e s o f A , B , C

s i n (l n x ) d xx

s i n 2 x

c o s 3 x d x

x l n (x + 1 ) d xM a t h e m a t i c s @ V i d y a l a n k a r I n s t i t u t e I n t e g r a t i o n - I n d e n i t e


17/40

P a r t i a l F r a c t i o n s

T o r e s o l v e a p r o p e r f r a c t i o n P /Q i n t o i t s s i m p l e s t s e t o f p a r t i a l f r a c t i o n s

1

T o a n o n - r e p e a t e d f a c t o r x

a o f Q t h e r e c o r r e s p o n d s a f r a c t i o n

a

o f t h e

f o r m

A

x a2

T o a f a c t o r ( x b )n o f Q t h e r e c o r r e s p o n d s a g r o u p o f t h e f o r m B

1

( x b ) +B

2

(x b ) 2 + +B

n

( x b ) n

3

T o a n o n - r e p e a t e d q u a d r a t i c f a c t o r x

2 + p x + q o f Q t h e r e c o r r e s p o n d s a

f r a c t i o n o f t h e f o r m

C x+

D

x

2 + p x + q

4

T o a f a c t o r ( x 2 + p x + q )n o f Q t h e r e c o r r e s p o n d s a g r o u p o f t h e f o r m

C

1

x

+D

1

x

2 + p x + q+

C

2

x

+D

2

(x 2 + p x + q ) 2+ +

C

n

x

+D

n

( x 2 + p x + q )n

w h e r e A , Bi

, Ci

, Di

a r e a l l i n d e p e n d e n t o f x

a

A f r a c t i o n P /Q w h e r e P , Q a r e p o l y n o m i a l s , i s t e r m e d p r o p e r f r a c t i o n i fd e g ( P ) < d e g ( Q ) e l s e i t s d e n e d a s i m p r o p e r f r a c t i o n



18/40

P a r t i a l F r a c t i o n s

E x a m p l e

1

U s e H e a v i s i d e C o v e r u p m e t h o d

a

a n d w r i t e

x

2 +1

( x

1 )( x

2 )( x

3 )

i n t o i t s p a r t i a l f r a c t i o n s f o r m

2

2 x + 7

(x

+1

)(x

2 +4

)d x i n t o p a r t i a l f r a c t i o n s f o r m

3

(x

a) (

x

b) (

x

c)

(x) (x ) ( x ) d x

a

B u t o n l y t r u e f o r l i n e a r f a c t o r w i t h n o r e p e t i t i o n



19/40

S p e c i a l s u b s t i t u t i o n

S u b s t i t u t i o n s

1

x

+1

x

2

=

x 1

x

2

+4

=x

2 +1

x

2

+2

2

d x +1

x = 1 1

x

2 d x3

d

x 1

x

=

1

+1

x

2

d x

1 x 2 + 1x

4

+ 1d x

2

x

2 1

x

4 +1

d x

3 x 2x

4

+ 1d x

4

1

x

4 +1

d x



20/40

Q u a d r a t i c i n t e g r a n d s

1

Q

d x

f a c t o r i z i n g a x

2

+b x

+c a n d t h e n t a k i n g h e l p f r o m p a r t i a l f r a c t i o n s

i f n o t f a c t o r i z a b l e t h e n f o r m p e r f e c t s q u a r e f o r m a t i n d e n o m i n a t o r a n d

u s e

1

x

2 +a 2d x o r

1

x

2 a

2

d x

1Q d xu s e

1

a

2 x

2

d x ,

1

x

2 a

2

d x ,

1

x

2 +a 2d x b y c o m p l e t i n g t h e s q u a r e

f o r m a t i n t h e d e n o m i n a t o r

Q d xu s e

a

2 x 2 d x , x 2 a 2 d x , x 2 + a 2 d x b y c o m p l e t i n g t h e s q u a r e f o r m a t i n s i d e t h e s q u a r e - r o o t

N o t e Q=

a x

2 +b x

+c



21/40

P r o b l e m s

1

1

2 x

2 + 8 x + 2 0d x

2

x

2 + 2 x + 5 d x

3 13 2 x x 2 d x

4

x

a

2 x

4

d x

5 x

x

4 +a

2

d x



22/40

Q u a d r a t i c I n t e g r a n d s

LQ

d x , L

Q

d x ,

L

Q d x

w h e r e L = p x + q a n d Q = a x 2 + b x + c

W r i t e L= d Qd x + a n d s o l v e t o n d a n d

L

Q

d x =Q +

Q

d x = l n | Q |+ Q

d x

L

Q

d x = Q +

Q

d x = 2

Q +1

Q

d x

L

Q d x =

(Q +)

Q d x =

Q

3 /2

3 /2+

Q d x



23/40

P r o b l e m s

1

x

+3

x

2 2 x

5

d x

2

5 x + 3x

2 +4 x

+1 0

d x

3

( x 5 ) x 2 + x d x




24/40

Q

1

Q

2

d x ,

Q

1Q

2

d x ,

Q

1

Q

2

d x

w h e r e Q

1

= a x 2 + b x + c a n d Q2

= p x 2 + q x + r

P u t Q

1

= Q2

+Q2

+ , s o l v e t o n d ,,. S o w e h a v e

Q

1

Q

2

d x =Q

2

+Q 2

+

Q

2

d x = + l n | Q2

|+ 1Q

2

d x

Q

1Q

2

d x=

Q

2

+Q

2

+ Q

2

d x=

Q

2

d x+

2

Q

2

+

1Q

2

d x

Q

1 Q 2 d x = (Q 2 +Q 2 + )Q 2 d x




25/40

1

1

L

1

L

2

d x ,

L

1L

2

d x ,

L

2

L

1

d x t h e n p u t L

2

= t 2

2

1

Q

L

d x t h e n p u t L = t 2

3

1

L

Q

d x t h e n p u t L=

1

t

N o t e

:

1

L

n

Q

d x : L=

1

t

4

1

Q

1

Q

2

d x t h e n p u t x =1

t

n o t e Q

i

a r e p u r e q u a d r a t i c i . e .

Q

i = a i x2

+ b i t h e n t2

= u


P r o b l e m s


26/40

1

1

( x 1 ) x 2 + 4 d x

2

( x 2 )

1 + x

1 x d x

3

1( 1 x 2 ) 1 + x 2 d x

4

1

(x

3)

x+

1

d x

5 1(

2 x

3)

4 x

x

2

d x




27/40

( x )

a x

2 + b x + cd x w h e r e ( x ) i s p o l y n o m i a l o f d e g r e e n

(

x)

a x

2 +b x

+c

d x = f (x )

a x

2 + b x + c +

1a x

2 +b x

+c

d x

D i e r e n t i a t e a n d

multiply a x2

+b x

+c

we get

( x ) = f (x )( a x 2 + b x + c ) +1

2

(2 a x + b )f ( x ) +

Note heref

is a polynomial of degreen

1

E x a m p l e

x

3 x 1x

2 +2 x

+2

d x

x

2

x

2 + 4 d x


R a t i o n a l i z a t i o n o f s p e c i a l a l g e b r a i c i r r a t i o n a l e x p r e s s i o n s


28/40

I f i n t e g r a n d i s a r a t i o n a l f u n c t i o n o f f r a c t i o n a l p o w e r s o f a n

i n d e p e n d e n t v a r i a b l e x , i . e . R

x,

x

p

1

/q1 ,

x

p

2

/q2 , . . . ,

x

p

k

/qk

t h e n t h e i n t e g r a n d c a n b e r a t i o n a l i z e d b y x = t m w h e r e m

=L C M

(q

1

,q

2

, . . . ,q

k

)

M o r e o v e r i f x i s r e p l a c e d b y

a x + b

c x+

d

i n t h e f u n c t i o n R( )

t h e n w e m a k e t h e s u b s t i t u t i o n

a x+

b

c x + d= t m w h e r e s a m e ,

m=

L C M(

q

1

,q

2

, . . . ,q

k

)


P r o b l e m s


29/40

E x a m p l e

1

x

+ 3x

4

x

5 6

x

7

d x ,

2 (

2 x

3) 1 /2

(2 x 3 )

1

/3

+1

d x

3

1

3

(

x+

1) 2 (

x

1) 4

d x

4 1x

2 + 3

x 1

x

d x


M i s c e l l a n e o u s P r o b l e m s


30/40

1

5 x

4 + 4 x 5

(x

5 +x

+1

)2d x

2

1 + x

2

1

x

2

1

1 + x 2 + x 4

d x


I n t e g r a t i o n o f a B i n o m i a l d i e r e n t i a l


31/40

x

m ( a + b x n ) p d x m , n , p Q

S o l v a b l e c a s e s

1

p Z+ t h e n w e e x p a n d (a + b x n ) u s i n g b i n o m i a l b u t i f p < 0t h e w e p u t x

=t

k

w h e r e k i s c o m m o n d e n o m i n a t o r o f t h e

f r a c t i o n s m a n d n

2

i f

m+

1

n

Zt h e n a

+b x

n =t

w h e r e i s d e n o m i n a t o r o f

f r a c t i o n p

3

i f

m + 1

n

+p Z t h e n a + b x n = t x n w h e r e i s d e n o m i n a t o r

o f f r a c t i o n p


P r o b l e m s


32/40

1

3 x 2 + x 2 d x

2

x

2 /3 1

+x

2 /3

1

d x

3

1 + 3

x

3

x

2

d x

4

x

1 1 1 + x 4

1

/2

d x

5

31

+ 4

xx

d x

6

1x

4

1 + x 2

d x


E u l e r s u b s t i t u t i o n


33/40

I n t e g r a l s o f t h e f o r m

R ( x , a x 2 + b x + c ) d x c a n b e e v a l u a t e d

u s i n g o n e o f t h e t h r e e e u l e r s u b s t i t u a t i o n s .

1

a x

2 + b x + c = t x a i f a > 02

a x

2 +b x

+c

=t x

c i f c>

0

3 a x 2 + b x + c = ( x ) t i f a x 2 + b x + c = a ( x ) (x )E x a m p l e

1

1

+x

2

+2 x

+2

d x


T r i g o n o m e t r i c s u b s t i t u t i o n


34/40

s i n

m

x c o s

n

x d x

1

i f n i s a n o d d n u m b e r , t h e s u b s t i t u t i o n s i n x

=t

2

i f m i s o d d , t h e s u b s t i t u t i o n c o s x = t

3

i f m+

n i s e v e n ,

1

ma n d

nb o t h n o n - n e g a t i v e e v e n n u m b e r s t h e n r e d u c e p o w e r

u s i n g c o s

2

x

=

1

+c o s 2

x

2

a n d s i n

2

x

=

1

c o s 2

x

2

2

e i t h e r o r b o t h o f m

a n dn

i s n e g a t i v e t h e n w e p u t t a n x = t

o r

c o tx = t


P r o b l e m s


35/40

E x a m p l e s

1 s i n 3 x3

c o s

2

x

d x p o w e r o f s i n i s o d d , t

=c o s x

2

c o s

3

x

s i n

3

x

d x m,

n n o t b o t h n o n - n e g a t i v e h e n c e t=

c o t x

3 s i n4

x c o s

6

x d x m , n e v e n , r e d u c e p o w e r b u t a l s o t h i n k !

4

s i n

2

x

c o s

6

x

d x t = t a n x

5

c o s

4

x

s i n

6

x

d x

6

1c o s

4

x

d x

7

1

3

s i n

1 1

x c o s x

d x

8 t a n7

x d x



36/40

T r i g o n o m e t r i c s u b s t i t u t i o n


37/40

1 a c o s x + b s i n xp c o s x + q s i n x

d x

,a e x + b e x

p e

x

+ q e xd x : N r

= D r

+(D r

)

2

1

a c o s

2

x + b s i n 2 xd x

,

1

a + b s i n 2 xd x

,

1

a+

b c o s

2

x

d x,

1

( a c o s x + b s i n x )2d x ,

1

a + b c o s 2 x + c s i n 2 x d x : Divide byc o s

2

xboth Nr & Dr,

t=

t a n x

3

1

a+

b c o s x

d x,

1

a+

b s i n x

d x,

1

a+

b c o s x+

c s i n x

d x : U s e

u n i v e r s a l s u b s t i t u t i o n

4

I f

p c o s x + q s i n x + ra c o s x

+b s i n x

+c

d x c a n b e s i m p l i e d b y w r i t i n g

N r = D r +d

d x

D r +


P r o b l e m s


38/40

E x a m p l e s

1

( U s i n g u n i v e r s a l s u b s t i t u t i o n )

1

1

s i nx (

2+

c o sx

2 s i n x )

d xU n i v e r s a l s u b s t i t u t i o n

2

1

5+

s i nx +

3 c o s x

d xU n i v e r s a l s u b s t i t u t i o n

2

1

s i n x ( 2 c o s 2 x 1 ) d x m u l t i , s i n ( x ) , p u t t = c o s x

3

s i n

2

x c o s x

s i n x + c o s xd x U n i v e r s a l s u b s t i t u t i o n

4

11 + c o t x

d x N r = D r +D r

5

1

2 s i n

2

x + 2 s i n x c o s x + c o s 2 xd x p u t t

=t a n x


R e d u c t i o n F o r m u l a


39/40

D e n i t i o n

A n y f o r m u l a w h i c h e x p r e s s e s a n i n t e g r a l i n t e r m s o f a n o t h e r i n t e g r a l w h i c h i s

s i m p l e r a n d i s o f t h e s a m e c l a s s a s t h e r s t , i s c a l l e d t h e r e d u c t i o n f o r m u l a f o r

t h e r s t i n t e g r a l

E x a m p l e s

1

I

n =

s i n

n

x d x

(n Z

+

)t h e n I

n = c o s x s i n

n

1

x

n

+n

1

n

I

n 2

1

H e n c e d e r i v e

s i n

6

x d xa n d

s i n

5

x d x

2

I

n

= c o sn

x d x(

n Z+) t h e n I

n

=s i n x c o s

n 1x

n

+n 1

n

I

n

2

3

I

n

=

t a n

n

x d x ( n Z+) t h e n In

=t a n

n 1x

n 1 I n 2

4

I

n

=

c o t

n

x d x(

n Z+) t h e n I

n

= c o t n 1 x

n

1

I

n

2


R e d u c t i o n f o r m u l a


40/40

E x a m p l e s

1

I

n

=

s e c

n

x d x(

n Z+)

t h e n I

n

=t a n s e c

n 2x

n

1

n 2n

1

I

n 2

2

I

n

= c s cn

x d x ( n Z+) t h e n In

=

c o t x c s c

n

2

x

n

1

+n

2

n

1

I

n 2

3

I

m , n =

s i n

m

x s i n

n

x d x (m , n Z+) t h e n I

m , n =s i n

m +1x c o s

n

1

x

m + n+

n

1

m + nI

m ,n 2

4

I

m , n = c o sm

x c o s n x d x ( m , n Z+) t h e n I

m

,n

=c o s

m

x s i n n x

n

+m

m + nI

m

1

,n

1


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