Evaluate the integral .

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

Evaluate the integral ∫dx12+4x−x2\displaystyle\int \frac{dx}{\sqrt{12+4x-x^{2}}}. (i) Prove that 12+4x−x2=16−(x−2)212+4x-x^{2}=16-(x-2)^{2}.​​ (ii) Using the result of part (i), determine the value of ∫dx12+4x−x2\displaystyle\int \frac{dx}{\sqrt{12+4x-x^{2}}}, writing your answer in its simplest form.

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