Civil Engineering (CE) Exam > Civil Engineering (CE) Notes > Engineering Mathematics > Integration
Integration | Engineering Mathematics - Civil Engineering (CE) PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
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
??
???
??
??
Read More
|
65 videos|120 docs|94 tests
|
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.
Related Exams
About this Document
|
4.92/5 Rating |
|
Dec 31, 2024 Last updated |
Document Description: Integration for Civil Engineering (CE) 2024 is part of Engineering Mathematics preparation.
The notes and questions for Integration have been prepared according to the Civil Engineering (CE) exam syllabus. Information about Integration covers topics
like and Integration Example, for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Integration.
Introduction of Integration in English is available as part of our Engineering Mathematics
for Civil Engineering (CE) & Integration in Hindi for Engineering Mathematics course.
Download more important topics related with notes, lectures and mock test series for Civil Engineering (CE)
Exam by signing up for free. Civil Engineering (CE): Integration | Engineering Mathematics - Civil Engineering (CE)
Description
Full syllabus notes, lecture & questions for Integration | Engineering Mathematics - Civil Engineering (CE) - Civil Engineering (CE) | Plus excerises question with solution to help you revise complete syllabus for Engineering Mathematics | Best notes, free PDF download
Information about Integration
In this doc you can find the meaning of Integration defined & explained in the simplest way possible. Besides explaining types of
Integration theory, EduRev gives you an ample number of questions to practice Integration tests, examples and also practice Civil Engineering (CE)
tests
|
Explore Courses for Civil Engineering (CE) exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
Integration | Engineering Mathematics - Civil Engineering (CE)
,study material
,Important questions
,MCQs
,Free
,Exam
,video lectures
,Previous Year Questions with Solutions
,Viva Questions
,Integration | Engineering Mathematics - Civil Engineering (CE)
,Extra Questions
,Integration | Engineering Mathematics - Civil Engineering (CE)
,practice quizzes
,mock tests for examination
,past year papers
,Sample Paper
,Summary
,Semester Notes
,Objective type Questions
,shortcuts and tricks
,ppt
;
Additional Information about Integration for Civil Engineering (CE) Preparation
Integration Free PDF Download
The Integration is an invaluable resource that delves deep into the core of the Civil Engineering (CE) exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Integration now and kickstart your journey towards success in the Civil Engineering (CE) exam.
Importance of Integration
The importance of Integration cannot be overstated, especially for Civil Engineering (CE) aspirants.
This document holds the key to success in the Civil Engineering (CE) exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Integration Notes
Integration Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Integration.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Integration Notes on EduRev are your ultimate resource for success.
Integration Civil Engineering (CE) Questions
The "Integration Civil Engineering (CE) Questions" guide is a valuable resource for all aspiring students preparing for the
Civil Engineering (CE) exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Integration on the App
Students of Civil Engineering (CE) can study Integration alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Integration,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Integration is prepared as per the latest Civil Engineering (CE) syllabus.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup