Indefinite and Definite Integration (Solved MCQs and Notes)

Notes 64 Pages

Joint Entrance Examination (Main)

Mathematics

Contributed by

Nikita Narasimhan

400
Unit - 9
Indefinite And Definite Integration
Important Points
1. If
 
 
( )
d
F x c f x
dx
 
then




f x dx F x c 

( )f x dx

is indefinite integral of f(x) w.r.to x where c is the arbitrary constant.
Rules of indefinite Integration
1 If f and g are integrable function on [a,b] and f+g is also integrable function on [a,b], then








( )f x g x dx f x dx g x dx  
  
.
If
1 2
, , . . . ,
n
f f f an integrable function on [a,b] then
       
 
     
1 2 3 1 2
... ... .
n n
f x f x f x f x dx f x dx f x dx f x dx       
   
2 (i) If f is integrable on [a, b] and k is the real constant then, kf is also integrable then





kf x dx k f x dx
 
(ii)
     
1 1 2 2
..........
n n
k f x k f x k f x dx   
 






1 1 2 2 n n
k f d k f d ......... k f x dx   
  
x x x x
3 If f and g are integrable functions on [a, b] then










f x g x dx f x dx g x dx  
  
 Important formulae
1
 
1
; 1 ;
1


     


n
n
x
x dx c n R x R
n
If n = 0 then
d c 

x x
2
 
1
d log| | c; R 0   

x x x
x
3 (i)
 
; 1 ,
log
x
x
e
a
a dx c a R x R
a

    

(ii)
e dx e c; R   

x x
x
4
sin cos ,
x dx x c x R
    

Page 1
401
5
cos sin ,
x dx x c x R
   

6
 
tan c , 2 k 1 , k Z
2

    x x x
7
Z  

2
cosec x dx = - cot x + c , x k , k
8
 
Z
π
s ec x ta n x d x = s e c x + c , x 2 k - 1 , k
2
 

9
,  

cosec x.cotxd x = -cosec x+c, x k k Z
10
 
1
2 2
1 1
tan , 0 ,
x
dx c a R x R
x a a a

    


 
1
1
cot , 0 ,

     
x
c a R x R
a a
11
 
2 2
dx 1 x - a
= log + c, a R - 0 , x ± a
x - a 2a x + a
 

12
 
2 2
dx 1 x+ a
= log + c , a R - 0 , x ± a
a - x 2a x - a
 

13
2
2
log , | | | |
dx
x x k c x k
x k
   


14
 
1
2 2
sin , , , 0
dx x
c x a a a
a
a x

    


 
1
cos , , ; 0
x
c x a a a
a

     
15
1
2 2
1 1
sec , | | | | 0
| |
x
dx c x a
a a
x x a

   



1
1
cos , | | | | 0
x
ec c x a
a a

    
16.
 
1
2
1 1
tan , , 0
b
dx x c a b
a bx a
ab

 
  
 
 

 

Method of substitution
* If


: ,
g R
 

is continuous and differentiable on
 
,
Page 2
402
and
 
tg
 is continuous and non zero on
 
,
if R
g

 
,a b
and
 
Rb,a:f 
is continuous and x = g(t) then
 
f x dx

=




[ ' ]f g t g t dt

* If




f x dx F x

+c then
   
1
f ax b dx F a b
a

  

+ C,
wherc
RI:f 
is continuous
 
0a 
*
   
 
   
 
1
'
, 1, 0, 0
1
n
n
f x
f x f x dx c n f x f x
n

 
 

     


*
 
 
 
f' x
dx = log | f x | + c,
f x

( f and
f

are continuous
   
f 0, f 0x x

 
)
*


 
 
'
2
f x
dx f x c
f x
 

( f and
f

are continuous
   
f 0, f 0x x

 
)
17.
tan x dx = log | sec x| + c

= - log | cos x | + c
,
2
k
x k z

 
18.
cot log| sin | ,
x dx x c
 

,
2
k
x k z

 
log | cosec | c  x
19.
cosec log|cosec cot | ,
x dx x x c
 

,
2
k
x k z

 
x
= log | tan | + c
2
20.
sec log | sec tan | , ,
2
k
x dx x x c x k z

    

log | tan | , ,
4 2 2
x k
c x k z
 
    
Integrals Substitutions
(i)
2 2
x a

tanx a 
or
cotx a 
(ii)
2 2
x a

secx a 
or
cosecx a 
(iii)
2 2
a x
sinx a 
or
cosx a 
Page 3
403
(iv)
a x
a x


cos 2x a 
(v)
2
2ax x
2
2 sinx a 
(vi)
 
2
2 2
2ax x a x a   
sinx a a  
or
cosa
For the integrals :
1 1
,
cos sina b x a c x 
and
1
,
cos sina b x c x 
taking
tan
2
x
t
* Integration by parts
du
u v dx u v dx v dx dx
dx
 
 
 
 
   
21.
2
2 2 2 2 2 2
log
2 2
x a
x a dx x a x x a c
      

22.
2
2 2 2 2 2 2
log
2 2
x a
x a dx x a x x a c
      

23.
 
2
2 2 2 2 1
sin 0
2 2
x a x
a x dx a x c a
a

     

24.
 
2 2
sin sin cos
ax
ax
e
e bx dx a bx b bx c
a b
  


25.
 

ax
ax
2 2
e
e cos bx dx = a cos bx+ b sin bx + c
a + b
26.
 
ax
ax
2 2
e
e sin bx dx = sin bx - θ + c
a + b

 
2 2
2 2
cos ; sin ; 0, 2
a b
a b
a b
      


27.
 
2 2
cos cos
ax
ax
e
e bx dx bx c
a b
   


 




 2,0;
ba
b
sin;
ba
a
cos
22
22
Page 4
404
28.








'  

x x
e f x f x dx e f x c
Definite Integration
Limit of a Sum
1.
0
1
( ) lim ( )
b
n
h
i
a
f x d x h f a ih


 


2.
 
1
lim
b
n
n
i
a
b a b a b a
f x dx f a i Where h
n n n


 
  
 
 
 
 
 
 


Fundamental theorem of definite Integration
If f is continuous on [a, b] and F is differentiable on (a, b) such that
( , ) ( ( )) ( )
d
x a b if F x f x then
dx
  
     
b
a
f x dx F b F a
 

Rules of definite Integration
1 If f and g are continuous in [a, b] then
 
( ) ( )
b
a
f x g x dx


=
( ) ( )
b
a
f x dx g x dx 

2 If f is continuous on [a, b] and k is real constant, then
   

 
b b
a a
k f n dx k f x dx
3 If f is continuous on the [a, b] and a < c < b then
( ) ( ) ( )
b c b
a a c
f x dx f x dx f x dx 
  
4
( ) ( )
b a
a b
f x dx f x dx 
 
5
( ) ( ) ( )
b b b
a a a
f x dx f t dt f u du 
  
Theorems
1 If f is even and continuous on the [-a, a] then
0
( ) 2 ( )
a a
a
f x dx f x dx


 
2 If f is odd and continuous on the [-a, a] then
( ) 0
a
a
f x dx



Page 5
405
3 If f is continuous on [0, a] then
0 0
( ) ( ) 
 
a a
f x dx f a x dx
4 If f is continuous on [a, b] then
   
b b
a a
f d f a b d  
 
x x x x
5 If f is continuous on [0, 2a] then
2
0 0 0
( ) ( ) (2 )
a a a
f x dx f x dx f a x dx  
  
Application of Integration
1 The area A of the region bounded by the curve y=f(x), X - axis and the lines
x = a, x = b is given by A =

I

, where I =
( )
b
a
f x dx

or I =
b
a
ydx

2 The area A of the region bounded by the curve x = g(y) and the line y = a and y = b given
by A =

I

Where I =
( )
b
a
g y dy

or I =
b
a
ydx

3 If the curve y = f(x) intersects X - axis at (c, 0) only and a < c < b then the area of the
region bounded by y = f(x), x = a, x = b and X - axis is given by
A =

I
1

+

I
2

where I
1
=
c
a
ydx

, I
2
=
b
c
ydx

4 If two curves y = f
1
(x) and y = f
2
(x) intersect each other at only two points for
x = a and x = b (a

b) then the area enclosed by them is given by
A =

I

and I =
 
1 2
( ) ( )
b
a
f x f x dx

5 If the two curves x = g
1
(y) and x = g
2
(y) intersect each other at only two points for
y = a and y = b (a

b) then the area enclosed by them is given by
A =

I

where I =
 
1 2
( ) ( )
b
a
g y g y dy

Page 6
406
Question Bank
(Indefinite Integration)
2
2 2
(1) ________
1 tan
(a) log sec tan (b) 2sec
2
1
(c) log sin (d) log sin cos
2
1
(2) ________
1
(a) 2log
x
x
x x
dx
c
x
x
x x
x x x x x
e
dx c
e
e e

 


 
  
 

 




2 2
5log 3log
4log 2log
2 3
(b) 2log
(c) 2log 1 (d) log 1
(3) _________
(a) 2 (b) log (c
x x
x x
x x
x x
x
e
e e
e e
e e
dx c
e e
e e x



 

 



 
3 2
) (d)
3 2
(4) _________
1
1 1 1
(a) log (b) log
1
1 1 1
(c) log 1 (d) log
log
(5)
n
n n
n n
n
n
n
x x
dx
c
x x
x x

n x n x
x
x
n n x
x
 







 
 
   
1
2 2
cot
2
1 log
_________
1
(a) log log 1 (b) log 1 log
1 1 1
(c) log (d) log
2
(6) 1 _________
1

x
x
dx c
x x
x x x x
x x
x x
x
e dx c
x


 

   
    
   
 
   
   
   
   
 
  
 

 


1 1 1 1
cot cot cot cot
1 1
(a) (b) (c) (d)
2 2
x x x x
xe e xe e
   
Page 7
407
 
   
1
4log 5
4 4
tan
(7) _________
cos
2 1 2 3
(a) (b) (c) (d)
cos cos 3 cos 2 cos
(8) 1 _________
1
(a) log 1 (b) - log 1 (c)
5
x
x
dx c
x
x x x x
e x c
x x

 
  

  
 


   
4 5
3
1
log 1 (d) log 1
5
(9) cos _________
1 1 1
(a) cos cot log cos cot (b) cos cot
2 2 2
1 1 1 1
(c) cos cot log cos cot (d) cos cot log cos cot
2 2 2 2
x x
ec x dx c
ec x x ecx x ec x x
ec x x ecx x ec x x ecx
 
 
   
   

 
2
2
1
1
3

2
(10) If 2 then _______
1 1
(a) (b) log 2 (c) 2 (d)
2log 2 2
(11) 1 _________
(a)
x
x
e
x
x
x
dx k c k
x
x e dx c
xe

  
   
  


   
 
     
     
(b) (c) (d)
(12) sin log cos log _______
(a) sin log cos log (b) sin log
(c) cos log (d) sin log cos log
(
x x x
xe xe xe
x x dx c
x x - x x
x x x x
 
 
  

 

  
             
 
7
9 8 8 8 8
1 1 2 1 1 2
13) 4 3 _________
3 3 3 8 33 3 8 33 3
(a) (b) (c) (d)
9 8 72 72 8
(14) _________
3 2
(a) 2 tan 2 (b) 2 tan 3 (c) 2 tan (d) 2 tan 2
(1
x x dx c
x x x x x x x
dx
c
x x
x x x x
   
   
      

 
 
  


       
-1 -1
2 2 2 2
5) _________
2
1 1
(a) 1 (b) 1 (c) 1 (d) 1
2 2
x
x x
x x x x
e
c
e e
e e e e

 
 
       

Page 8
408
2
cos
(16) If log sin 1 then _________
sin 2sin 1
1
(a) 2 (b) 1 (c) (d) -2
2
(17) _________
1

x
x
dx A x c A
x x
dx
c
e
   
 
 



2
8 8
2 2
1 1
(a) log (b) log (c) log (d) log
1 2 1
cos sin
(18) ________
1 2sin cos
cos2 sin 2
(a) - (b) -
2
x x x x
x x x x
e e e e
e e e e
x x
dx c
x x
x x
 
 
 

 


 
 
 
 
2
1
1
1 1
cos 2 sin 2
(c) (d)
2 2 2
1
(19) log _________
1 log
tan log
tan
(a) (b) tan log (c) (d) tan
(20) I
x x
d x dx c
x
x
x x
x x


 
 


 
2
1 cos8x
f cos8 then A _______
cot 2 tan 2
1 1 1 1
(a) (b) (c) (d)
16 8 16 8
4 6
(21) If log 9 4 th
9 4
x x
x
x x
dx A x C
x x
e e
dx Ax B e c
e e



  

 

   



1
6 6
en ______ and _____
3 35 3 35 3 35 3 35
(a) (b) (c) (d)
2 36 2 36 2 36 2 36
tan 2
(22) If tan then _______
cos 2
1
(a)
2
A B
, , , ,
dx x
K c K
sm x x

 
   
 
  
 

 

3
2
3
2
1
(b) 1 (c) 1 (d)
2
(23) If 1 then _______
1
4 3
(a) (b)
3 4
x
dx P x c P
x
 
   


 
-4 -3
(c) (d)
3 4
sec
(24) __________
sin 2 sin
xdx
c
x
 
  

Page 9
409
   
   
4
1 1 3
6
(a) 2sec tan tan (b) 2sec tan tan
(c) 2sec cot cot (d) 2sec cot cot
1
(25) If tan tan then _______
1
(a) 3
x x
x x
x
dx x P x c P
x
 
     
     

   


 
2
1 1
(b) (c) (d) 3
3 3
log 1
(26) _________
log
(a) log (b) log (c) (d)
log log
(27)
x
dx c
x
x x
x x x x
x x
 

 



 
 
2
log
________
(a) log (b) log (c) log (d) log
(28) If cosec cot log sin then _______
(a) 1
x x
x
x x x x x
e ex
dx c
x
e
x e x e x x xe
x
x xdx P x x Q x c P Q
 
     


6 7 7
(b) 2 (c) 0 (d) -1
(29) If log log then _________
6 1 1
(a) (b) (c)
49 49 49
x xdx Px x Qx c P Q    


 
 
     
6
(d)
49
1
(30) log log _________
log
1
(a) (b) log log (c) log log (d) log log
log log
x dx c
x
x
x x x x x
x x

 
  
 
 
 

2
2
1
2
2
1
(31) __________
1 1 1 1
(a) (b) (c) (d)
x
x
x
e
x
dx c
x x x x
x x x x
e e e e

 

 
 
 
    

Page 10

Indefinite and Definite Integration (Solved MCQs and Notes)

Indefinite and Definite Integration (Solved MCQs and Notes)

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