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Mathematics
Problem
Explain why the function defined by
F
(
x
)
=
y
0
+
∫
a
x
f
(
t
)
d
t
F(x)=y_0+\int_{a}^{x}f(t)\,dt
F
(
x
)
=
y
0
+
∫
a
x
f
(
t
)
d
t
necessarily satisfies both the differential equation
d
F
d
x
=
f
(
x
)
\dfrac{dF}{dx}=f(x)
d
x
d
F
=
f
(
x
)
and the initial condition
F
(
a
)
=
y
0
F(a)=y_0
F
(
a
)
=
y
0
. Your answer should refer to one calculus theorem that justifies the result.
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