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Mathematics
Mathematics
Problem
For a positive parameter
a
a
a
, define
I
(
a
)
=
∫
0
a
2
x
sin
(
x
2
+
a
)
d
x
.
I(a)=\displaystyle \int_{0}^{\sqrt{a}} 2x\,\sin\bigl(x^{2}+a\bigr)\,dx.
I
(
a
)
=
∫
0
a
2
x
sin
(
x
2
+
a
)
d
x
.
a. (i) Evaluate
I
(
a
)
I(a)
I
(
a
)
in terms of
a
a
a
. Clearly show each step of your working. b. (ii) Using the expression obtained in part (i), find the exact value of
I
(
π
2
)
I\!\bigl(\tfrac{\pi}{2}\bigr)
I
(
2
π
)
.\
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