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Mathematics
Mathematics
Problem
Let
f
(
x
)
=
{
sin
x
,
x
≤
π
2
x
−
1
,
x
>
π
2
f(x)=\begin{cases}\sin x,& x\le \frac{\pi}{2}\\x-1,& x> \frac{\pi}{2}\end{cases}
f
(
x
)
=
{
sin
x
,
x
−
1
,
x
≤
2
π
x
>
2
π
and define
F
(
x
)
=
∫
0
x
f
(
t
)
d
t
F(x)=\int_{0}^{x} f(t)\,dt
F
(
x
)
=
∫
0
x
f
(
t
)
d
t
. Given that
F
(
0
)
=
0
F(0)=0
F
(
0
)
=
0
, write an explicit piecewise formula for
F
(
x
)
F(x)
F
(
x
)
valid for all real
x
x
x
.
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