integration [dx(x² +1)/√x²+2 is equal to
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Calculus Examples
Popular Problems Calculus Find the Integral 1/(x^2-1)
1
x
2
−
1
Factor the numerator and denominator of
1
x
2
−
1
.
∫
1
(
x
+
1
)
(
x
−
1
)
d
x
Write the fraction using partial fraction decomposition.
∫
A
1
x
+
1
+
A
2
x
−
1
d
x
Simplify.
∫
−
1
2
(
x
+
1
)
+
1
2
(
x
−
1
)
d
x
Split the single integral into multiple integrals.
∫
−
1
2
(
x
+
1
)
d
x
+
∫
1
2
(
x
−
1
)
d
x
Since
−
1
is constant with respect to
x
, move
−
1
out of the integral.
−
∫
1
2
(
x
+
1
)
d
x
+
∫
1
2
(
x
−
1
)
d
x
Since
1
2
is constant with respect to
x
, move
1
2
out of the integral.
−
(
1
2
∫
1
x
+
1
d
x
)
+
∫
1
2
(
x
−
1
)
d
x
Let
u
1
=
x
+
1
. Then
d
u
1
=
d
x
. Rewrite using
u
1
and
d
u
1
.
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−
1
2
∫
1
u
1
d
u
1
+
∫
1
2
(
x
−
1
)
d
x
The integral of
1
u
1
with respect to
u
1
is
ln
(
|
u
1
|
)
.
−
1
2
(
ln
(
|
u
1
|
)
+
C
)
+
∫
1
2
(
x
−
1
)
d
x
Since
1
2
is constant with respect to
x
, move
1
2
out of the integral.
−
1
2
(
ln
(
|
u
1
|
)
+
C
)
+
1
2
∫
1
x
−
1
d
x
Let
u
2
=
x
−
1
. Then
d
u
2
=
d
x
. Rewrite using
u
2
and
d
u
2
.
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−
1
2
(
ln
(
|
u
1
|
)
+
C
)
+
1
2
∫
1
u
2
d
u
2
The integral of
1
u
2
with respect to
u
2
is
ln
(
|
u
2
|
)
.
−
1
2
(
ln
(
|
u
1
|
)
+
C
)
+
1
2
(
ln
(
|
u
2
|
)
+
C
)
Simplify.
−
1
2
ln
(
|
u
1
|
)
+
1
2
ln
(
|
u
2
|
)
+
C
Substitute back in for each integration substitution variable.
.
−
1
2
ln
(
|
x
+
1
|
)
+
1
2
ln
(
|
x
−
1
|
)
+
C