Consider the functions f, g, defined for x∈R.

Question

Consider the functions f, g, defined for x∈R. given by f(x)=e^-x sin x and g(x)= e^-x cos x

(a) Find

i. f'(x)

iI. g'(x)

(b) Hence, or otherwise, find ∫ e^-x sin x dx , between π and 0.

Answer 1

d(e^-x sin x )/dx=e^-xcos x +(sinx)(- e^-x )=e^-x(cosx-sinx)

d(e^-x cos x)/dx= -e^-xsinx +(-e^-x)cosx =-e^-x(sinx +cosx)

∫ e^-x sin x dx= e^-x(sinx -cosx)/2 for [0, π] =(e^-π +1)/2