ENTEGRALS
BANPLE
2
Evaluate
Eraluate
the
following
integrals:
LAMPLE
S
Evaluate.
2
1+
cos
2x
dx
BASED
ON
BASIC
CONCEPTS
(BASIC)
DXAMPLE
4
Evaluate:
4+1
1
e
ILLUSTRATIVE
EXAMPLES
2
2
cos
x
Sloge
_p4
loge
x
3
loge
2
loge
x
+C
=
+1
1
3+1
-3+1
-c-2
+c
SOLLTION
We
know
that
eo8e
X=
2
+1
Sloge
x4
loge
X
5
cos
x
-+C
=
2xV2
+C
3
loge
X _2loge X
2
cos
x
2
+C
+C=--
+C
2x2
1
dx
4
x3+1
3+1
dx
()
We
know
that
cos
2x
= 2 cosx-1.
+C=
SOLUTION
(0)
We
know
that
1--cos
2x
=2
sin
x.
Cos
2x
+ 2
sinx
4
SOLUTION
We
know
that
1 +
cos
2x
=2
cos
x
and
1
-
cos
2x
=
2
sinx.
Therefore,
2 2
1+
cos
2x
2
cos
2x
+2
sin
x
dv
INCERT,
CBSE
2018
(1)
cos
+C
(x-1)
u-1)
(G0)
)1-cos
2x
2
dx
=cosec
x dx= -
cot
x
+C
2
sin
x
dx
=sec
x
dx
=
tan
x
+C
cos
x
dx
[Using
formula
(1)]
[Using
formula
(1)]
[Using
formula
()]
[Using
formula
(1)]
2
cos
x-cos
2x
15.3
sin
x
-(l-2sin"
x+2
sin
x
dy=
[
dx
=
see
x
dx
=
tan
x
+C
3
dx
+C
