Question 1

Paper 2, calculatorHard6 marks

For a real parameter k>0k>0 let

fk(x)=x2−kx2+1,−3≤x≤3.f_k(x)=\frac{x^2-k}{x^2+1},\qquad -3\leq x\leq 3.

It is known that

∫−33fk(x) dx=0.\int_{-3}^{3} f_k(x)\,dx=0.
(a)

Determine the exact value of kk

[2]
(b)

Hence determine the total area between the graph of y=fk(x)y=f_k(x) and the xx-axis for −3≤x≤3-3\leq x\leq 3 Show a definite-integral expression before evaluating. Do not use numerical integration. Give your answer to 3 significant figures.

[4]

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