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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
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O u t l i n e
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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 )
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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 )
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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
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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
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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
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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
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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
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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
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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
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E x t e n s i o n o f f o r m u l a e f r o m b a s i c
E x t e n d e d f r o m e l e m e n t a r y
1
t a n x d x = l n | s e c x |+ c
2
c o t x d x = l n | s i n x |+ c
3
s e c d x = l n | s e c x + t a n x |+ c = l n t a n x
2
+
4 + c4
c o s e c d x = l n | c o s e c x c o t x |+ c = l n | t a n ( x /2 ) |+ c
5
1
x
2 +a
2
d x=
1
a
t a n
1 xa
+c
6 1
x
2
a2
d x =1
2 a
l n x ax+
a + c7
1
a
2 x
2
d x= 1
x
2 a
2
d x
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Q u a d r a t i c I n t e g r a n d s
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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
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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
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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
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( 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
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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
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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
)
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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
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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
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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
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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
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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
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E u l e r s u b s t i t u t i o n
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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
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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
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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
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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
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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
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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 +
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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
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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
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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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