Consider the family of functions fn(x)=cosnx, where x∈R and n∈N.
(a) By writing cosnx as cosn−1xcosx, show that
∫cosnxdx=cosn−1xsinx+(n−1)∫cosn−2xdx−(n−1)∫cosnxdx for n>1.
(b) Hence, show that ∫fn(x)dx=n1cosn−1xsinx+nn−1∫fn−2(x)dx for n>1.
(c) Hence, find an expression for ∫cos4xdx, giving your answer in the form pcos3xsinx+qcosxsinx+rx+c where p,q,r∈Q+.
The region R is enclosed by the graph of y=cos2x and the x-axis where −2π≤x≤2π, as shown in the following diagram.
The region R is rotated by 2π radians around the x-axis to form a solid of revolution.
(d) Find the volume of the solid.
(e.i) Find the Maclaurin series of fn(x) up to the term in x2.
(e.ii) Hence or otherwise, find limx→0x2fn(x)−1 in terms of n.