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Mathematics
Mathematics
Problem
Answer the questions below. (i) Derive general formulae for
∫
sin
(
a
x
+
b
)
d
x
\displaystyle \int \sin(ax+b)\,dx
∫
sin
(
a
x
+
b
)
d
x
and
∫
cos
(
a
x
+
b
)
d
x
\displaystyle \int \cos(ax+b)\,dx
∫
cos
(
a
x
+
b
)
d
x
where
a
≠
0
a\ne 0
a
=
0
. (ii) Using your result, evaluate
I
(
a
)
=
∫
0
π
sin
(
a
x
)
d
x
I(a)=\displaystyle \int_{0}^{\pi}\sin(ax)\,dx
I
(
a
)
=
∫
0
π
sin
(
a
x
)
d
x
and simplify the answer. (iii) Hence, find
lim
a
→
0
I
(
a
)
\displaystyle \lim_{a\to 0} I(a)
a
→
0
lim
I
(
a
)
.
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