Q.The value of $\int \sqrt{e^x}\,dx$ is :

$\int\frac{\sin 2x}{2\cos x}\,dx=\int\sin x\,dx=-\cos x+c$.

Using the identity $\sin 2x=2\sin x\cos x$,

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Simplify the integrand with the double-angle identity $\sin 2x=2\sin x\cos x$; it collapses to $\int\cos x\,dx=\sin x+c$.

Start with the given integral:

$\int\frac{\sin 2x}{2\sin x}\,dx.$

Apply $\sin 2x=2\sin x\cos x$: …

Convert the surd to an exponent, $\sqrt{e^x}=e^{x/2}$, then use $\int e^{ax}dx=\frac{1}{a}e^{ax}$ with $a=\tfrac12$.

Rewrite the integrand:

$\int\sqrt{e^x}\,dx=\int e^{x/2}\,dx.$

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