Methods2025 NHT Exam 1 · Q8Tech-free6 marks

Integration by recognition of a surd function

Let $f: [0, \infty) \to \R,\ f(x) = \dfrac{x}{2\sqrt{x+k}}$ , where is a positive real number.

a.i.

Consider the particular case where $k = 6$ .

Show that $\dfrac{d}{dx}\left(x\sqrt{x+6}\right) = \sqrt{x+6} + \dfrac{x}{2\sqrt{x+6}}$ .

1 mark

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a.ii.

Hence, or otherwise, evaluate the definite integral $\displaystyle\int_0^6 f(x)\,dx$ .

3 marks

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Find $\displaystyle\int_0^k f(x)\,dx$ in the form $\dfrac{a-\sqrt{a}}{b}k\sqrt{k}$ where $a$ and $b$ are positive integers.

2 marks

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