
12-6 Calculus – Differentiation and Integration
=
∫
∫
= +
− −
−
cos (cos ),
(cos )
.
1 1
1 2
2
x d x
x dx
x
c
which is in the form
b.
which is
e
x
dx e
dx
x
e d x
x
x
x
tan
tan
tan
(tan ),
−
−
−
+
∫
=
+
∫
=
∫
−
1
1
1
1 1
2 2
1
in the form e dx
e c
x
x
∫
= +
−
tan
.
1
Example 12.10
a. ( )( ) ,x x x dx
2
1 2 1+ + +
∫
b. csc .x dx
ò
Solution:
a.
( )( ) ( ) ( ),x x x dx x x d x x
x dx
2 2 2
1 2 1 1 1+ + +
∫
= + + + +
∫
∫
=
where is in the form
(( )
.
x x
c
2 2
1
+ +
+
b. csc
sin
sin cos
sec
tan
x dx
x
dx
x x
dx
x
x
∫
=
∫
=
∫
=
1 1
2
2 2
1
2 2
2
2
ddx
x
x
dx
x
d
x
∫
=
∫
=
1
2
1
2 2
1
2
2
2
tan
sec
tan
tan
∫ ∫
=
+
,
log tan
which is in the form
1
2
x
dx
x
cc x x c= − +log(csc cot ) .