Oxford University MAT 2017 Solved Papers

Considering each of the integrals in turn:
(a) note that (x2−4)sin8(πx)≤0 when 0≤x≤2 so this integral will evaluate as negative;
(b) (2+cos‌x) will always evaluate positive as it translates the y=cos‌x graph positively along the y-axis, so the integral will be positive and will be less than 54π for 0≤x≤2π;
(c) the maximum value of sin100(x) is 1 , and so the value of the integral for 0≤x≤π must be ≤π;
(d) for 0≤x≤π the smallest (3−sin‌x)6 can be is 64π;
(e) (sin3(x)−1) will always be ≤0, and so this integral will evaluate as negative.