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Integration/Substitution/dx version/Fact/Proof
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Integration/Substitution/dx version/Fact
Proof
Because of
fact
, we have
∫
φ
−
1
(
a
)
φ
−
1
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b
)
f
(
φ
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s
)
)
φ
′
(
s
)
d
s
=
∫
φ
(
φ
−
1
(
a
)
)
φ
(
φ
−
1
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b
)
)
f
(
t
)
d
t
=
∫
a
b
f
(
t
)
d
t
.
{\displaystyle {}\int _{\varphi ^{-1}(a)}^{\varphi ^{-1}(b)}f(\varphi (s))\varphi '(s)\,ds=\int _{\varphi {\left(\varphi ^{-1}(a)\right)}}^{\varphi {\left(\varphi ^{-1}(b)\right)}}f(t)\,dt=\int _{a}^{b}f(t)\,dt\,.}
To fact
