Maths
Calculus - Integration
∫
u
d
v
d
x
d
x
=
u
v
-
∫
d
u
d
x
v
d
x
∫
x
n
d
x
,
where
n
&neq;
-
1
=
x
n
+
1
n
+
1
+
c
∫
1
x
d
x
=
ln
|
x
|
+
c
∫
cos
a
x
d
x
=
1
a
sin
a
x
+
c
∫
sin
a
x
d
x
=
-
1
a
cos
a
x
+
c
∫
tan
x
d
x
=
ln
|
sec
x
|
+
c
∫
csc
x
d
x
=
ln
|
tan
x
|
+
c
∫
sec
x
d
x
=
ln
|
tan
(
¼π
+
x
)
|
+
c
∫
cot
x
d
x
=
ln
|
sin
x
|
+
c
∫
1
x
2
d
x
=
tanh
-1
x
+
c
∫
d
x
x
2
+
a
2
=
sinh
-1
(
x
a
)
+
c
∫
1
1
-
x
2
d
x
=
sin
-1
x
+
c
∫
d
x
x
2
-
a
2
=
cosh
-1
(
x
a
)
+
c
,
x
>
a
>
0
∫
cosh
x
d
x
=
sinh
x
+
c
∫
sinh
x
d
x
=
cosh
x
+
c
∫
tanh
x
d
x
=
ln
|
cosh
x
|
+
c
∫
coth
x
d
x
=
ln
|
sinh
x
|
+
c
∫
sech
2
x
d
x
=
tanh
x
+
c
∫
csch
2
x
d
x
=
-
coth
x
+
c
∫
sech
x
tanh
x
d
x
=
-
sech
x
+
c
∫
csch
x
coth
x
d
x
=
-
csch
x
+
c
f
′
(
x
)
f
(
x
)
=
|
f
(
x
)
|
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