) Let f(x) = 2x - x². Prove that f is integrable on [0,2] using either the definition or the - characterisation of integrability. Hint: you may use the following formula's without proof:e п(n + 1) k = n(n + 1)(2n + 1) %3D 2 6. k=1 k=1

Q5 C7

Let A and u be positive real numbers. Then Jp(Ax) and Jp(ux) satisfy d d + dx dx (11.6.37) d ( d p² dx dx (11.6.38) respectively. Show that for 2 # µ, | xJp(ax)Jp(ux)dx (11.6.39) 12 - µ2 [Hint: Multiply (11.6.37) by Jp(ux), (11.6.38) by Jp(Ax), subtract the resulting equations and integrate over (0, 1).] If A and u are distinct zeros of Jp(x), what does your result imply?

Ⓒ Prove that f(x)= x² + 2x² is integraliśle of integrals. on [1,1]] using definition