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Mathematics
Mathematics
Problem
Define
P
(
x
)
=
∫
0
x
sin
(
2
t
2
+
1
)
d
t
P(x)=\displaystyle \int_{0}^{x}\sin(2t^{2}+1)\,dt
P
(
x
)
=
∫
0
x
sin
(
2
t
2
+
1
)
d
t
. (i) Find
P
′
(
x
)
P'(x)
P
′
(
x
)
. (ii) Show that
P
′
′
(
x
)
=
4
x
cos
(
2
x
2
+
1
)
P''(x)=4x\cos(2x^{2}+1)
P
′′
(
x
)
=
4
x
cos
(
2
x
2
+
1
)
. (iii) Hence, determine
P
′
′
(
π
/
2
)
P''(\sqrt{\pi/2})
P
′′
(
π
/2
)
.
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