Write a value

given $\int e^{\log \sin x} \cos x d x$

$=\int \sin x \cos x d x\left(\because e^{\log x}=x\right)$

Let $\sin x=t$

Differentiating on both sides we get,

$\cos x d x=d t$

Substituting above equations in given equation we get,

$=\int \mathrm{t} \mathrm{dt}$

$=\frac{t^{2}}{2}+c$

$=\frac{\sin ^{2} x}{2}+c$