When would you use u substitution twice?
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Answer:
When we are reversing a differentiation that had the composition of three functions. Here is one example.
Explanation:
∫
sin
4
(
7
x
)
cos
(
7
x
)
d
x
Let
u
=
7
x
. This makes
d
u
=
7
d
x
and our integral can be rewritten:
1
7
∫
sin
4
u
cos
u
d
u
=
1
7
∫
(
sin
u
)
4
cos
u
d
u
To avoid using
u
to mean two different things in one discussion, we'll use another variable (
t
,
v
,
w
are all popular choices)
Let
w
=
sin
u
, so we have
d
w
=
cos
u
d
u
and our integral becomes:
1
7
∫
w
4
d
w
We the integrate and back-substitute:
1
7
∫
w
4
d
w
=
1
35
w
5
+
C
=
1
35
sin
5
u
+
C
=
1
35
sin
5
7
x
+
C
If we check the answer by differentiating, we'll use the chain rule twice.
d
d
x
(
(
sin
(
7
x
)
)
5
)
=
5
(
sin
(
7
x
)
)
4
⋅
d
d
x
(
sin
(
7
x
)
)
=
5
(
sin
(
7
x
)
)
4
⋅
cos
(
7
x
)
d
d
x
(
7
x
)
=
5
(
sin
(
7
x
)
)
4
⋅
cos
(
7
x
)
⋅
7
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