\(\text{Integration}\)

\(\int_0^{\pi/2}4^{\sin x}\cos xdx\)

\(\text{Let us take a substitution}\)

\(\sin x=z\)

\(\cos x dx=dz\)

\(\int_0^1 4^zdz\)

\(=[\frac{4^z}{\ln 4}]_0^1=\frac{4}{\ln4}-\frac{1}{\ln4}\)

\(=\frac{3}{\ln4}\)

\(\text{This is the required answer.}\)

\(\int_0^{\pi/2}4^{\sin x}\cos xdx\)

\(\text{Let us take a substitution}\)

\(\sin x=z\)

\(\cos x dx=dz\)

\(\int_0^1 4^zdz\)

\(=[\frac{4^z}{\ln 4}]_0^1=\frac{4}{\ln4}-\frac{1}{\ln4}\)

\(=\frac{3}{\ln4}\)

\(\text{This is the required answer.}\)