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Mathematics
Mathematics
Problem
Consider the function
g
(
x
)
=
6
sin
(
x
+
π
6
)
+
5
(
x
−
3
)
2
+
4
−
9
cos
x
.
g(x)=6\sin\bigl(x+\tfrac{\pi}{6}\bigr)+\frac{5}{(x-3)^2+4}-9\cos x.
g
(
x
)
=
6
sin
(
x
+
6
π
)
+
(
x
−
3
)
2
+
4
5
−
9
cos
x
.
(i) Find a general antiderivative
G
(
x
)
G(x)
G
(
x
)
of
g
(
x
)
g(x)
g
(
x
)
. (ii) Show by direct differentiation that
d
d
x
G
(
x
)
=
g
(
x
)
\dfrac{d}{dx}G(x)=g(x)
d
x
d
G
(
x
)
=
g
(
x
)
. (iii) Using your expression for
G
(
x
)
G(x)
G
(
x
)
, evaluate the exact value of
G
(
π
3
)
−
G
(
0
)
G\bigl(\tfrac{\pi}{3}\bigr)-G(0)
G
(
3
π
)
−
G
(
0
)
.
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