Answer
$A = {b^3}{e^{b\sin br}}$
Work Step by Step
$$\eqalign{
& \frac{{dA}}{{dr}} = A{b^2}\cos br,{\text{ }}A\left( 0 \right) = {b^3} \cr
& {\text{Separating the variables}} \cr
& \frac{{dA}}{A} = {b^2}\cos brdr \cr
& {\text{Integrate both sides of the equation}} \cr
& \int {\frac{{dA}}{A}} = \int {A{b^2}\cos brdr} \cr
& \int {\frac{{dA}}{A}} = b\int {\cos br\left( b \right)dr} \cr
& \ln \left| A \right| = b\sin br + C{\text{ }}\left( {\bf{1}} \right) \cr
& {\text{Use the initial condition }}A\left( 0 \right) = {b^3} \cr
& \ln \left| {{b^3}} \right| = b\sin b\left( 0 \right) + C \cr
& C = \ln \left| {{b^3}} \right| \cr
& {\text{Substitute }}C{\text{ into }}\left( {\bf{1}} \right) \cr
& \ln \left| A \right| = b\sin br + \ln \left| {{b^3}} \right| \cr
& {\text{Solve for }}A \cr
& {e^{\ln \left| A \right|}} = {e^{b\sin br + \ln \left| {{b^3}} \right|}} \cr
& A = {e^{\ln \left| {{b^3}} \right|}}{e^{b\sin br}} \cr
& A = {b^3}{e^{b\sin br}} \cr} $$