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 Page 1


    
Integration 
Integrand and element of integration 
The function under the sign of integration is called integrand. For e.g. in
3
xdx
?
; x
3
 is 
called integrand. In the integral f(x) dx
?
, dx is known as the element of integration and it 
indicates the variable with respect to which the given function is to be integrated. 
 
Constant of integration : 
We know that  
  
22
d
(x ) 2x 2xdx x
dx
??
?
 
Also 
2
d
(x c) 2x
dx
?? ? 
2
2xdx x c ? ?
?
, where c is any constant 
So we notice that x
2 
is an integral of 2x, then x
2
 + c is also an integral of 2x. In general if  
  f(x)dx
?
  =   ?(x)  then 
  f(x)dx
?
  =   ?(x) + c 
 
Standard formulae 
n1
n
x
xdx c
n1
?
??
?
?
,  (n ? ?1) 
1
22
1x
sin c
a
ax
?
? ?
?
?
 
1
dx logx c
x
??
?
 
1
22
dx x
cos c
a
ax
?
???
?
?
 
xx
edx e c ??
?
 
1
22
dx 1 x
tan c
aa
ax
?
? ?
?
?
 
x
x
e
a
adx
log a
?
?
 + c 
1
22
dx 1 x
cot c
aa
ax
?
?
? ?
?
?
 
cosxdx sinx c ??
?
 
1
22
dx 1 x
sec c
aa
xx a
?
? ?
?
?
 
cosecxcotxdx cosecx c ???
?
 
1
22
dx 1 x
cosec c
aa
xx a
?
?
? ?
?
?
 
sinxdx cosx c ?? ?
?
 sinhxdx
?
  =  cos h x + c 
Page 2


    
Integration 
Integrand and element of integration 
The function under the sign of integration is called integrand. For e.g. in
3
xdx
?
; x
3
 is 
called integrand. In the integral f(x) dx
?
, dx is known as the element of integration and it 
indicates the variable with respect to which the given function is to be integrated. 
 
Constant of integration : 
We know that  
  
22
d
(x ) 2x 2xdx x
dx
??
?
 
Also 
2
d
(x c) 2x
dx
?? ? 
2
2xdx x c ? ?
?
, where c is any constant 
So we notice that x
2 
is an integral of 2x, then x
2
 + c is also an integral of 2x. In general if  
  f(x)dx
?
  =   ?(x)  then 
  f(x)dx
?
  =   ?(x) + c 
 
Standard formulae 
n1
n
x
xdx c
n1
?
??
?
?
,  (n ? ?1) 
1
22
1x
sin c
a
ax
?
? ?
?
?
 
1
dx logx c
x
??
?
 
1
22
dx x
cos c
a
ax
?
???
?
?
 
xx
edx e c ??
?
 
1
22
dx 1 x
tan c
aa
ax
?
? ?
?
?
 
x
x
e
a
adx
log a
?
?
 + c 
1
22
dx 1 x
cot c
aa
ax
?
?
? ?
?
?
 
cosxdx sinx c ??
?
 
1
22
dx 1 x
sec c
aa
xx a
?
? ?
?
?
 
cosecxcotxdx cosecx c ???
?
 
1
22
dx 1 x
cosec c
aa
xx a
?
?
? ?
?
?
 
sinxdx cosx c ?? ?
?
 sinhxdx
?
  =  cos h x + c 
   
secx tanxdx secx c ??
?
 coshxdx
?
= sin h x + c 
2
sec x dx tanx c ??
?
 
2
dx
x1 ?
?
= sin h
?1
 x + c 
2
cosec xdx
?
= ?cot x + c 
2
dx
x1 ?
?
=  cos h
?1
 x + c 
cotxdx
?
 = log sin x + c 
2
dx
x1 ?
?
  =  tan h
?1
 x + c 
tanxdx
?
 = log sec x + c  tanxdx
?
 = ? log (cos x) + c 
? ? secxdx log(secx tanx) c ?? ?
?
 secxdx ?
?
? ??
x
log tan c
42
???
? ?
??
??
 
?? ? ? cosec x dx log(cosecx cotx) c ?? ? ?
?
 ? ? cosec x dx ?
?
x
log tan
2
??
??
??
 + c 
 
Important Trigonometric Identities 
? sin
2
 A + cos
2
 A =1 
? sin (A + B) = sin A cos B + cos A sin B 
? cos (A + B) = cos A cos B ? sin A sin B 
? tan (A + B)  = 
tanA tanB
1 tanA tanB
?
?
 
? sin (A ? B)  = sin A cos B ? cos A sin B 
 
? cos (A ? B)  = cos A cos B + sin A sin B 
? tan (A ? B)  = 
tanA tanB
1 tanA tanB
?
?
 
? sin
2
 A ? sin
2
 B   =  sin (A + B) sin (A ? B) 
 
? cos
2
 A ? sin
2
 B  =  cos (A + B) cos (A ? B) 
 
? 2 sin A cos B = sin (A + B) + sin (A ? B) 
 
? 2 cos A sin B  =  sin (A + B) ? sin (A ? B) 
 
? 2 cos A cos B  =  cos (A + B) + cos (A ? B) 
Page 3


    
Integration 
Integrand and element of integration 
The function under the sign of integration is called integrand. For e.g. in
3
xdx
?
; x
3
 is 
called integrand. In the integral f(x) dx
?
, dx is known as the element of integration and it 
indicates the variable with respect to which the given function is to be integrated. 
 
Constant of integration : 
We know that  
  
22
d
(x ) 2x 2xdx x
dx
??
?
 
Also 
2
d
(x c) 2x
dx
?? ? 
2
2xdx x c ? ?
?
, where c is any constant 
So we notice that x
2 
is an integral of 2x, then x
2
 + c is also an integral of 2x. In general if  
  f(x)dx
?
  =   ?(x)  then 
  f(x)dx
?
  =   ?(x) + c 
 
Standard formulae 
n1
n
x
xdx c
n1
?
??
?
?
,  (n ? ?1) 
1
22
1x
sin c
a
ax
?
? ?
?
?
 
1
dx logx c
x
??
?
 
1
22
dx x
cos c
a
ax
?
???
?
?
 
xx
edx e c ??
?
 
1
22
dx 1 x
tan c
aa
ax
?
? ?
?
?
 
x
x
e
a
adx
log a
?
?
 + c 
1
22
dx 1 x
cot c
aa
ax
?
?
? ?
?
?
 
cosxdx sinx c ??
?
 
1
22
dx 1 x
sec c
aa
xx a
?
? ?
?
?
 
cosecxcotxdx cosecx c ???
?
 
1
22
dx 1 x
cosec c
aa
xx a
?
?
? ?
?
?
 
sinxdx cosx c ?? ?
?
 sinhxdx
?
  =  cos h x + c 
   
secx tanxdx secx c ??
?
 coshxdx
?
= sin h x + c 
2
sec x dx tanx c ??
?
 
2
dx
x1 ?
?
= sin h
?1
 x + c 
2
cosec xdx
?
= ?cot x + c 
2
dx
x1 ?
?
=  cos h
?1
 x + c 
cotxdx
?
 = log sin x + c 
2
dx
x1 ?
?
  =  tan h
?1
 x + c 
tanxdx
?
 = log sec x + c  tanxdx
?
 = ? log (cos x) + c 
? ? secxdx log(secx tanx) c ?? ?
?
 secxdx ?
?
? ??
x
log tan c
42
???
? ?
??
??
 
?? ? ? cosec x dx log(cosecx cotx) c ?? ? ?
?
 ? ? cosec x dx ?
?
x
log tan
2
??
??
??
 + c 
 
Important Trigonometric Identities 
? sin
2
 A + cos
2
 A =1 
? sin (A + B) = sin A cos B + cos A sin B 
? cos (A + B) = cos A cos B ? sin A sin B 
? tan (A + B)  = 
tanA tanB
1 tanA tanB
?
?
 
? sin (A ? B)  = sin A cos B ? cos A sin B 
 
? cos (A ? B)  = cos A cos B + sin A sin B 
? tan (A ? B)  = 
tanA tanB
1 tanA tanB
?
?
 
? sin
2
 A ? sin
2
 B   =  sin (A + B) sin (A ? B) 
 
? cos
2
 A ? sin
2
 B  =  cos (A + B) cos (A ? B) 
 
? 2 sin A cos B = sin (A + B) + sin (A ? B) 
 
? 2 cos A sin B  =  sin (A + B) ? sin (A ? B) 
 
? 2 cos A cos B  =  cos (A + B) + cos (A ? B) 
    
? 2 sin A sin B   =  cos (A ? B) ? cos (A + B) 
? 2 sin 
CD
2
?
 cos
CD
2
?
 = sin C + sin D 
? 2 cos 
CD
2
?
 sin 
CD
2
?
 = sin C ? sin D 
? 2 cos
CD
2
?
 cos
CD
2
?
 = cos C + cos D 
? 2 sin 
CD
2
?
 sin 
DC
2
?
 = cos C ? cos D 
 
? cos 2A = cos
2
 A ? sin
2
 A = 1 ? 2sin
2
 A = 2 cos
2
A ? 1 = 
2
2
1tan A
1tan A
?
?
 
? sin 2A  = 2 sin A cos A = 
2
2tanA
1tan A ?
 
 
? tan 2A  =  
2
2tanA
1tan A ?
 
 
? sin 3A = 3 sin A ? 4 sin
3
A 
 
? cos 3A =  4 cos
3 
A ? 3 cos A 
? tan 3A  = 
3
2
3tanA 4tan A
13tan A
?
?
 
 
 
Note :  Integration of  
mn
sin xcos xdx
?
 where m and n are positive integers 
(i) If m be odd and n be even, for integration put t = cos x 
(ii) If m be even and n be odd, for integration put t = sin x 
(iii) If m and n are odd, for integration put either t = cos x or sin x 
(iv) If m and n are even, for integration put either t = cos x or sin x 
 
Solved Example 17 : 
Evaluate
3
sin x dx
?
        
Solution :  
sin 3x = 3 sin x ? 4 sin
3
x 
4 sin
3
x = 3 sin x ? sin 3x 
?   4 
3
sin x dx
?
  = ??
?
(3sinx sin3x) dx 
 ? ??
??
3sinxdx sin3xdx 
  
3
1cos3x
sin xdx 3cosx c
43
??
? ?? ?
??
??
?
 
 
Page 4


    
Integration 
Integrand and element of integration 
The function under the sign of integration is called integrand. For e.g. in
3
xdx
?
; x
3
 is 
called integrand. In the integral f(x) dx
?
, dx is known as the element of integration and it 
indicates the variable with respect to which the given function is to be integrated. 
 
Constant of integration : 
We know that  
  
22
d
(x ) 2x 2xdx x
dx
??
?
 
Also 
2
d
(x c) 2x
dx
?? ? 
2
2xdx x c ? ?
?
, where c is any constant 
So we notice that x
2 
is an integral of 2x, then x
2
 + c is also an integral of 2x. In general if  
  f(x)dx
?
  =   ?(x)  then 
  f(x)dx
?
  =   ?(x) + c 
 
Standard formulae 
n1
n
x
xdx c
n1
?
??
?
?
,  (n ? ?1) 
1
22
1x
sin c
a
ax
?
? ?
?
?
 
1
dx logx c
x
??
?
 
1
22
dx x
cos c
a
ax
?
???
?
?
 
xx
edx e c ??
?
 
1
22
dx 1 x
tan c
aa
ax
?
? ?
?
?
 
x
x
e
a
adx
log a
?
?
 + c 
1
22
dx 1 x
cot c
aa
ax
?
?
? ?
?
?
 
cosxdx sinx c ??
?
 
1
22
dx 1 x
sec c
aa
xx a
?
? ?
?
?
 
cosecxcotxdx cosecx c ???
?
 
1
22
dx 1 x
cosec c
aa
xx a
?
?
? ?
?
?
 
sinxdx cosx c ?? ?
?
 sinhxdx
?
  =  cos h x + c 
   
secx tanxdx secx c ??
?
 coshxdx
?
= sin h x + c 
2
sec x dx tanx c ??
?
 
2
dx
x1 ?
?
= sin h
?1
 x + c 
2
cosec xdx
?
= ?cot x + c 
2
dx
x1 ?
?
=  cos h
?1
 x + c 
cotxdx
?
 = log sin x + c 
2
dx
x1 ?
?
  =  tan h
?1
 x + c 
tanxdx
?
 = log sec x + c  tanxdx
?
 = ? log (cos x) + c 
? ? secxdx log(secx tanx) c ?? ?
?
 secxdx ?
?
? ??
x
log tan c
42
???
? ?
??
??
 
?? ? ? cosec x dx log(cosecx cotx) c ?? ? ?
?
 ? ? cosec x dx ?
?
x
log tan
2
??
??
??
 + c 
 
Important Trigonometric Identities 
? sin
2
 A + cos
2
 A =1 
? sin (A + B) = sin A cos B + cos A sin B 
? cos (A + B) = cos A cos B ? sin A sin B 
? tan (A + B)  = 
tanA tanB
1 tanA tanB
?
?
 
? sin (A ? B)  = sin A cos B ? cos A sin B 
 
? cos (A ? B)  = cos A cos B + sin A sin B 
? tan (A ? B)  = 
tanA tanB
1 tanA tanB
?
?
 
? sin
2
 A ? sin
2
 B   =  sin (A + B) sin (A ? B) 
 
? cos
2
 A ? sin
2
 B  =  cos (A + B) cos (A ? B) 
 
? 2 sin A cos B = sin (A + B) + sin (A ? B) 
 
? 2 cos A sin B  =  sin (A + B) ? sin (A ? B) 
 
? 2 cos A cos B  =  cos (A + B) + cos (A ? B) 
    
? 2 sin A sin B   =  cos (A ? B) ? cos (A + B) 
? 2 sin 
CD
2
?
 cos
CD
2
?
 = sin C + sin D 
? 2 cos 
CD
2
?
 sin 
CD
2
?
 = sin C ? sin D 
? 2 cos
CD
2
?
 cos
CD
2
?
 = cos C + cos D 
? 2 sin 
CD
2
?
 sin 
DC
2
?
 = cos C ? cos D 
 
? cos 2A = cos
2
 A ? sin
2
 A = 1 ? 2sin
2
 A = 2 cos
2
A ? 1 = 
2
2
1tan A
1tan A
?
?
 
? sin 2A  = 2 sin A cos A = 
2
2tanA
1tan A ?
 
 
? tan 2A  =  
2
2tanA
1tan A ?
 
 
? sin 3A = 3 sin A ? 4 sin
3
A 
 
? cos 3A =  4 cos
3 
A ? 3 cos A 
? tan 3A  = 
3
2
3tanA 4tan A
13tan A
?
?
 
 
 
Note :  Integration of  
mn
sin xcos xdx
?
 where m and n are positive integers 
(i) If m be odd and n be even, for integration put t = cos x 
(ii) If m be even and n be odd, for integration put t = sin x 
(iii) If m and n are odd, for integration put either t = cos x or sin x 
(iv) If m and n are even, for integration put either t = cos x or sin x 
 
Solved Example 17 : 
Evaluate
3
sin x dx
?
        
Solution :  
sin 3x = 3 sin x ? 4 sin
3
x 
4 sin
3
x = 3 sin x ? sin 3x 
?   4 
3
sin x dx
?
  = ??
?
(3sinx sin3x) dx 
 ? ??
??
3sinxdx sin3xdx 
  
3
1cos3x
sin xdx 3cosx c
43
??
? ?? ?
??
??
?
 
 
   
Solved Example 18 : 
Evaluate  sin3x cos2x dx
?
  
Solution :  
1
sin3xcos2xdx (sin5x sinx)dx
2
??
??
 
      = 
11
cos5x cos x
10 2
?
? 
 
Solved Example 19 : 
Evaluate sin2xsin3xdx
?
 
Solution : 
?? ? ? ?
??
1
sin2x sin3x dx cos x cos5x dx
2
  
 = 
11
sinx sin5x
210
? 
 
Solved Example 20 : 
Integrate  
22
dx
cos xsin x
?
 
Solution :  
Now,  
?
?
22
22 22
1cosxsinx
cos x sin x cos x sin x
 
     ??
22
11
sin x cos x
 
     = cosec
2
 x + sec
2
x      
[ ?1 = sin
2 
x + cos
2
 x] 
? 
22
dx
cos xsin x
?
  
 = 
22
(cosec x sec x)dx ?
?
 
  = 
22
cosec xdx sec xdx ?
??
 
  = ? cot x + tan x + c 
 
Solved Example 21 :    
Evaluate 
33
sin x cos xdx
?
 
Solution :  
We have 
 sin
3
 x cos
3
 x  = (sin x cos x)
3    
    = 
1
8
(2 sin x cos x)
3   
      
= 
1
8
sin
3
 2x 
      = 
11
84
?  (3 sin 2x ? sin 6x)
   [ ? 4 sin
3
x = 3 sin x ? sin 3x] 
     = 
1
32
 (3 sin 2x ? sin 6x) 
? 
?
33
sin x cos xdx 
??
?
1
(3sin2x sin6x)dx
32
 
 = 
31
sin2xdx sin6xdx
32 32
?
??
 
 = 
3 cos2x 1 cos6x
c
32 2 32 6
?? ?? ??
? ?
?? ??
?? ??
 
Hence ?
?
33
sin x cos x dx 
?
? ??
31
cos2x cos6x c
64 192
 
 
Solved Example 22 : 
Evaluate 
1
dx
1sinx ?
?
 
Solution :  
111sinx
1 sinx 1 sinx 1 sinx
?
??
?? ?
 
    = 
?
?
2
1sinx
1sin x
 
Page 5


    
Integration 
Integrand and element of integration 
The function under the sign of integration is called integrand. For e.g. in
3
xdx
?
; x
3
 is 
called integrand. In the integral f(x) dx
?
, dx is known as the element of integration and it 
indicates the variable with respect to which the given function is to be integrated. 
 
Constant of integration : 
We know that  
  
22
d
(x ) 2x 2xdx x
dx
??
?
 
Also 
2
d
(x c) 2x
dx
?? ? 
2
2xdx x c ? ?
?
, where c is any constant 
So we notice that x
2 
is an integral of 2x, then x
2
 + c is also an integral of 2x. In general if  
  f(x)dx
?
  =   ?(x)  then 
  f(x)dx
?
  =   ?(x) + c 
 
Standard formulae 
n1
n
x
xdx c
n1
?
??
?
?
,  (n ? ?1) 
1
22
1x
sin c
a
ax
?
? ?
?
?
 
1
dx logx c
x
??
?
 
1
22
dx x
cos c
a
ax
?
???
?
?
 
xx
edx e c ??
?
 
1
22
dx 1 x
tan c
aa
ax
?
? ?
?
?
 
x
x
e
a
adx
log a
?
?
 + c 
1
22
dx 1 x
cot c
aa
ax
?
?
? ?
?
?
 
cosxdx sinx c ??
?
 
1
22
dx 1 x
sec c
aa
xx a
?
? ?
?
?
 
cosecxcotxdx cosecx c ???
?
 
1
22
dx 1 x
cosec c
aa
xx a
?
?
? ?
?
?
 
sinxdx cosx c ?? ?
?
 sinhxdx
?
  =  cos h x + c 
   
secx tanxdx secx c ??
?
 coshxdx
?
= sin h x + c 
2
sec x dx tanx c ??
?
 
2
dx
x1 ?
?
= sin h
?1
 x + c 
2
cosec xdx
?
= ?cot x + c 
2
dx
x1 ?
?
=  cos h
?1
 x + c 
cotxdx
?
 = log sin x + c 
2
dx
x1 ?
?
  =  tan h
?1
 x + c 
tanxdx
?
 = log sec x + c  tanxdx
?
 = ? log (cos x) + c 
? ? secxdx log(secx tanx) c ?? ?
?
 secxdx ?
?
? ??
x
log tan c
42
???
? ?
??
??
 
?? ? ? cosec x dx log(cosecx cotx) c ?? ? ?
?
 ? ? cosec x dx ?
?
x
log tan
2
??
??
??
 + c 
 
Important Trigonometric Identities 
? sin
2
 A + cos
2
 A =1 
? sin (A + B) = sin A cos B + cos A sin B 
? cos (A + B) = cos A cos B ? sin A sin B 
? tan (A + B)  = 
tanA tanB
1 tanA tanB
?
?
 
? sin (A ? B)  = sin A cos B ? cos A sin B 
 
? cos (A ? B)  = cos A cos B + sin A sin B 
? tan (A ? B)  = 
tanA tanB
1 tanA tanB
?
?
 
? sin
2
 A ? sin
2
 B   =  sin (A + B) sin (A ? B) 
 
? cos
2
 A ? sin
2
 B  =  cos (A + B) cos (A ? B) 
 
? 2 sin A cos B = sin (A + B) + sin (A ? B) 
 
? 2 cos A sin B  =  sin (A + B) ? sin (A ? B) 
 
? 2 cos A cos B  =  cos (A + B) + cos (A ? B) 
    
? 2 sin A sin B   =  cos (A ? B) ? cos (A + B) 
? 2 sin 
CD
2
?
 cos
CD
2
?
 = sin C + sin D 
? 2 cos 
CD
2
?
 sin 
CD
2
?
 = sin C ? sin D 
? 2 cos
CD
2
?
 cos
CD
2
?
 = cos C + cos D 
? 2 sin 
CD
2
?
 sin 
DC
2
?
 = cos C ? cos D 
 
? cos 2A = cos
2
 A ? sin
2
 A = 1 ? 2sin
2
 A = 2 cos
2
A ? 1 = 
2
2
1tan A
1tan A
?
?
 
? sin 2A  = 2 sin A cos A = 
2
2tanA
1tan A ?
 
 
? tan 2A  =  
2
2tanA
1tan A ?
 
 
? sin 3A = 3 sin A ? 4 sin
3
A 
 
? cos 3A =  4 cos
3 
A ? 3 cos A 
? tan 3A  = 
3
2
3tanA 4tan A
13tan A
?
?
 
 
 
Note :  Integration of  
mn
sin xcos xdx
?
 where m and n are positive integers 
(i) If m be odd and n be even, for integration put t = cos x 
(ii) If m be even and n be odd, for integration put t = sin x 
(iii) If m and n are odd, for integration put either t = cos x or sin x 
(iv) If m and n are even, for integration put either t = cos x or sin x 
 
Solved Example 17 : 
Evaluate
3
sin x dx
?
        
Solution :  
sin 3x = 3 sin x ? 4 sin
3
x 
4 sin
3
x = 3 sin x ? sin 3x 
?   4 
3
sin x dx
?
  = ??
?
(3sinx sin3x) dx 
 ? ??
??
3sinxdx sin3xdx 
  
3
1cos3x
sin xdx 3cosx c
43
??
? ?? ?
??
??
?
 
 
   
Solved Example 18 : 
Evaluate  sin3x cos2x dx
?
  
Solution :  
1
sin3xcos2xdx (sin5x sinx)dx
2
??
??
 
      = 
11
cos5x cos x
10 2
?
? 
 
Solved Example 19 : 
Evaluate sin2xsin3xdx
?
 
Solution : 
?? ? ? ?
??
1
sin2x sin3x dx cos x cos5x dx
2
  
 = 
11
sinx sin5x
210
? 
 
Solved Example 20 : 
Integrate  
22
dx
cos xsin x
?
 
Solution :  
Now,  
?
?
22
22 22
1cosxsinx
cos x sin x cos x sin x
 
     ??
22
11
sin x cos x
 
     = cosec
2
 x + sec
2
x      
[ ?1 = sin
2 
x + cos
2
 x] 
? 
22
dx
cos xsin x
?
  
 = 
22
(cosec x sec x)dx ?
?
 
  = 
22
cosec xdx sec xdx ?
??
 
  = ? cot x + tan x + c 
 
Solved Example 21 :    
Evaluate 
33
sin x cos xdx
?
 
Solution :  
We have 
 sin
3
 x cos
3
 x  = (sin x cos x)
3    
    = 
1
8
(2 sin x cos x)
3   
      
= 
1
8
sin
3
 2x 
      = 
11
84
?  (3 sin 2x ? sin 6x)
   [ ? 4 sin
3
x = 3 sin x ? sin 3x] 
     = 
1
32
 (3 sin 2x ? sin 6x) 
? 
?
33
sin x cos xdx 
??
?
1
(3sin2x sin6x)dx
32
 
 = 
31
sin2xdx sin6xdx
32 32
?
??
 
 = 
3 cos2x 1 cos6x
c
32 2 32 6
?? ?? ??
? ?
?? ??
?? ??
 
Hence ?
?
33
sin x cos x dx 
?
? ??
31
cos2x cos6x c
64 192
 
 
Solved Example 22 : 
Evaluate 
1
dx
1sinx ?
?
 
Solution :  
111sinx
1 sinx 1 sinx 1 sinx
?
??
?? ?
 
    = 
?
?
2
1sinx
1sin x
 
    
  
?
?? ?
22 2
1sinx 1 sinx
cosx cosx cosx
 
or 
1
1sinx ?
 = sec
2
x ? tan x sec x 
?     
?
?
1
dx
1sinx
 
??
?
2
(sec x tanxsec x)dx 
  = ?? ?
??
2
sec x dx tanx sec x dx 
?? ? tanx sec x c 
 
 
Solved Example 23 : 
Evaluate 
76
sin x cos xdx
?
 
Solution : 
?
?
76
sin x cos x dx 
?
?
66
sin x cos x sinxdx 
 = 
23 6
(1 cos x) cos xsinxdx ?
?
 
= ?? ?
?
24 6
(1 3cos x 3cos x cos x) 
    
6
cos x sinxdx 
= ?? ?
?
68 10 12
(cos x 3cos x 3cos x cos x) 
   sinxdx 
= 
6 8 10 12
(t 3t 3t t )dt ?? ? ?
?
 
by putting t = cos x 
= ?
7 9 11 13
tt t t
33
7 9 11 13
?? ? 
= ? ??
79 11
11 3
cos x cos x cos x
73 11
 
   ?
13
1
cos x
13
 
 
Solved Example 24 : 
Evaluate 
63
sin x cos xdx
?
 
Solution :  
?? ?
??
63 6 2
sin x cos x dx sin x(1 sin x)cos xdx
 
          = 
68
(sin x sin x)cos x dx ?
?
 
     = 
68
(t t )dt ?
?
    
? by putting  
    sin x = t = 
79
11
tt
79
? = 
79
11
sin x sin x
79
? 
 
Solved Example 25 : 
Evaluate 
42
sin x cos xdx
?
 
Solution :  
sin
4 
x cos
2 
x 
=  
1
8
(2 sin
2
x)
2
 (1+cos2x) 
= 
1
8
 (1 ?cos2x ? cos
2
2x + cos
3
 2x) 
= 
1
8
? ?
??
?
?
1cos4x
1cos2x
2
 
? ?
?
?
?
cos6x 3cos2x
4
 
= 
1
8
11 1 1
cos2x cos4x cos6x
24 2 4
??
???
??
??
 
= 
11 1
1cos2x cos4x cos6x
16 2 2
??
?? ?
??
??
 
?
42
sin x cos xdx 
= 
??
?? ?
??
??
11 1
1 cos2x cos4x cos6x dx
16 2 2
 
= 
11 1 1
xsin2x sin4x sin6x
16 4 4 12
??
???
??
??
 
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65 videos|120 docs|94 tests

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FAQs on Integration - Engineering Mathematics - Civil Engineering (CE)

1. What is integration in mathematics?
Ans. Integration in mathematics is a concept that involves finding the area under a curve. It is the reverse process of differentiation and is used to calculate quantities such as area, volume, displacement, and more.
2. How is integration used in real life?
Ans. Integration is used in various real-life applications such as calculating the area under a curve, determining the volume of irregular shapes, analyzing population growth, predicting future trends in economics, and more.
3. What are the different methods of integration?
Ans. Some common methods of integration include integration by substitution, integration by parts, trigonometric integration, partial fractions, and numerical integration techniques such as the trapezoidal rule or Simpson's rule.
4. What is the fundamental theorem of calculus related to integration?
Ans. The fundamental theorem of calculus states that differentiation and integration are inverse processes of each other. It establishes a connection between the concept of integration and differentiation, providing a method to evaluate definite integrals.
5. How can integration be used to solve problems in physics?
Ans. Integration is widely used in physics to calculate quantities such as work, energy, momentum, and more. It helps in analyzing motion, determining the center of mass, finding the gravitational potential energy, and solving differential equations that model physical systems.
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