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Mathematics
Problem
Evaluate the integral
∫
d
x
12
+
4
x
−
x
2
\displaystyle\int \frac{dx}{\sqrt{12+4x-x^{2}}}
∫
12
+
4
x
−
x
2
d
x
. a. (i) Prove that
12
+
4
x
−
x
2
=
16
−
(
x
−
2
)
2
12+4x-x^{2}=16-(x-2)^{2}
12
+
4
x
−
x
2
=
16
−
(
x
−
2
)
2
. b. (ii) Using the result of part (i), determine the value of
∫
d
x
12
+
4
x
−
x
2
\displaystyle\int \frac{dx}{\sqrt{12+4x-x^{2}}}
∫
12
+
4
x
−
x
2
d
x
, writing your answer in its simplest form.
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