find the integral of 1 / 4-x^2
Answers
Answer:
Evaluate integral of 1/(4-x^2) with respect to x
∫
1
4
−
x
2
d
x
Factor the numerator and denominator of
1
4
−
x
2
.
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∫
1
(
2
+
x
)
(
2
−
x
)
d
x
Write the fraction using partial fraction decomposition.
∫
A
1
2
+
x
+
A
2
2
−
x
d
x
Simplify.
∫
1
4
(
2
+
x
)
+
1
4
(
2
−
x
)
d
x
Split the single integral into multiple integrals.
∫
1
4
(
2
+
x
)
d
x
+
∫
1
4
(
2
−
x
)
d
x
Since
1
4
is constant with respect to
x
, move
1
4
out of the integral.
1
4
∫
1
2
+
x
d
x
+
∫
1
4
(
2
−
x
)
d
x
Let
u
1
=
2
+
x
. Then
d
u
1
=
d
x
. Rewrite using
u
1
and
d
u
1
.
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1
4
∫
1
u
1
d
u
1
+
∫
1
4
(
2
−
x
)
d
x
The integral of
1
u
1
with respect to
u
1
is
ln
(
|
u
1
|
)
.
1
4
(
ln
(
|
u
1
|
)
+
C
)
+
∫
1
4
(
2
−
x
)
d
x
Since
1
4
is constant with respect to
x
, move
1
4
out of the integral.
1
4
(
ln
(
|
u
1
|
)
+
C
)
+
1
4
∫
1
2
−
x
d
x
Let
u
2
=
2
−
x
. Then
d
u
2
=
−
d
x
, so
−
d
u
2
=
d
x
. Rewrite using
u
2
and
d
u
2
.
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1
4
(
ln
(
|
u
1
|
)
+
C
)
+
1
4
∫
−
1
u
2
d
u
2
Move the negative in front of the fraction.
1
4
(
ln
(
|
u
1
|
)
+
C
)
+
1
4
∫
−
1
u
2
d
u
2
Since
−
1
is constant with respect to
u
2
, move
−
1
out of the integral.
1
4
(
ln
(
|
u
1
|
)
+
C
)
+
1
4
(
−
∫
1
u
2
d
u
2
)
Simplify.
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1
4
(
ln
(
|
u
1
|
)
+
C
)
−
1
4
∫
1
u
2
d
u
2
The integral of
1
u
2
with respect to
u
2
is
ln
(
|
u
2
|
)
.
1
4
(
ln
(
|
u
1
|
)
+
C
)
−
1
4
(
ln
(
|
u
2
|
)
+
C
)
Simplify.
1
4
ln
(
|
u
1
|
)
−
1
4
ln
(
|
u
2
|
)
+
C
Substitute back in for each integration substitution variable.
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1
4
ln
(
|
2
+
x
|
)
−
1
4
ln
(
|
2
−
x
|
)
+
C
Hope,it is helpful