# Evaluate the following integrals. int_0^{pi/2}4^{sin x}cos xdx

Question
Integrals
Evaluate the following integrals.
$$\int_0^{\pi/2}4^{\sin x}\cos xdx$$

2021-02-07
$$\text{Integration}$$
$$\int_0^{\pi/2}4^{\sin x}\cos xdx$$
$$\text{Let us take a substitution}$$
$$\sin x=z$$
$$\cos x dx=dz$$
$$\int_0^1 4^zdz$$
$$=[\frac{4^z}{\ln 4}]_0^1=\frac{4}{\ln4}-\frac{1}{\ln4}$$
$$=\frac{3}{\ln4}$$
$$\text{This is the required answer.}$$

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