A model for the energy output, E(x)E(x)E(x), of a specialized transducer under mechanical stress is defined by the function E(x)=(3+4x)4E(x) = (3 + 4x)^4E(x)=(3+4x)4, where xxx is the applied strain. The expansion of this function is given by
(3+4x)4=A+Bx+864x2+768x3+256x4(3 + 4x)^4 = A + Bx + 864x^2 + 768x^3 + 256x^4(3+4x)4=A+Bx+864x2+768x3+256x4
Find the values of the constants AAA and BBB.
Show that
(3+4x)4−(3−4x)4=Cx+Dx3(3 + 4x)^4 - (3 - 4x)^4 = Cx + Dx^3(3+4x)4−(3−4x)4=Cx+Dx3
where CCC and DDD are constants to be found.
Hence, or otherwise, find
∫((3+4x)4−(3−4x)4)dx\int \left( (3 + 4x)^4 - (3 - 4x)^4 \right) dx∫((3+4x)4−(3−4x)4)dx